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JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

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Q.1. If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then the length of its latus rectum is    (2020)
(1) √3
(2) 3√2
(3) 3/√2
(4) 2√3
Ans. 
(2)
Solution. The distance between the foci of an ellipse is
2ae = 6 ⇒ ae = 3 ...(1)
The distance between ellipse directrices is
2a/e = 12 ⇒ a = 6e ...(2)
On solving Eqs. (1) and (2), we get
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Now, JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Hence, the length of latus rectum is
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.2. If y = mx + 4 is a tangent to both the parabolas y2 = 4x and x2 = 2by, then b is equal to   (2020)
(1) −32
(2) −64
(3) −128
(4) 128
Ans.
(3)
Solution. The equation of tangent of parabola y= 4ax is given by
y = mx + a/m ...(1)
Now, y = mx + 4 is a tangent to the parabola y2 = 4x, then
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Therefore, the equation of tangent is y = 4x + 1/4. It is also the tangent of parabola x2 = 2by, then
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
The value of b cannot be zero, hence b = −128.

Q.3. If 3x + 4y = 12√2 is a tangent to the ellipse JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE for some JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEEthen the distance between the foci of the ellipse is    (2020)
(1) 2√7
(2) 4
(3) 2√5
(4) 2√2
Ans.
(1)
Solution. The given tangent of ellipse JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEEis
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
The line y = mx + c will be a tangent of ellipse JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE if
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Now, the foci of ellipse is JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Hence, the distance between the foci of the ellipse is
2ae = 2 x 4 x √7/4 = 2√7

Q.4. Let the line y = mx and the ellipse 2x2 + y2 = 1 intersect at a point P in the first quadrant. If the normal to this ellipse at P meets the co-ordinate axes at JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE and (0, β), then β is equal to    (2020)
(1) (2√2)/3
(2) 2/(√3)
(3) 2/3
(4) (√2)/3
Ans.
(2)
Solution. Let the coordinates of point P be (x1, y1). So, the equation of normal at point P is
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE...(1)
It passes through the point JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE then
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Now, JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE (since P lies in first quadrant)
The normal also passes through the point(0, β), then
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.5. The locus of a point which divides the line segment joining the point (0, −1) and a point on the parabola x2 = 4y, internally in the ratio 1 : 2 is    (2020)
(1) 9x2 - 12y = 8
(2) 9x2 - 3y = 2
(3) x2 - 3y = 2
(4) 4x2 - 3y = 2
Ans.
(1)
Solution. Let the point on parabola x2 = 4y be (2t, t2). Therefore,
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Now,
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE ...(1)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE ...(2)
From Eqs. (1) and (2), we get
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.6. If a hyperbola passes through the point P (10, 16) and it has vertices at (±6, 0), then the equation of the normal to it at P is    (2020)
(1) 3x + 4y = 94
(2) 2x + 5y = 100
(3) x + 2y = 42
(4) x + 3y = 58
Ans.
(2)
Solution. Let the equation of hyperbola is
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
The coordinates of vertices are (±a, 0) = (±6, 0) ⇒ a = 6.
The equation of hyperbola is
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Now, hyperbola passes through the point P (10, 16), then
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Therefore, the equation of hyperbola is
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Now, the equation of normal at point P is
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.7. If a line y = mx + c is a tangent to the circle (x - 3)2 + y2 = 1 and it is perpendicular to a line L1, where L1 is the tangent to the circle x2 + y2 = 1 at the point JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE then    (2020)
(1) c2 - 7c + 6 = 0
(2) c2 + 7c + 6 = 0
(3) c2 + 6c + 7 = 0
(d) c2 - 6c + 7 = 0
Ans. 
(3)
Solution.

JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE


Q.8. If e1 and e2 are the eccentricities of the ellipse JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE and the hyperbola JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE respectively and (e1, e2) is a point on the ellipse 15x+ 3y2 = k, then k is equal to    (2020)
(1) 16
(2) 17
(3) 15
(4) 14
Ans.
(1)
Solution. The eccentricity of ellipse JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEEis
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE ...(1)
The eccentricity of hyperbola JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE is
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE ...(2)
Point (e1, e2) lies on the ellipse 15x2 + 3y2 = k, then
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.9. The length of the minor axis (along y-axis) of an ellipse in the standard form is 4/(√3). If this ellipse touches the line x + 6y = 8; then its eccentricity is    (2020)
(1) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
(2) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
(3) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
(4) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Ans.
(1)
Solution. The length of the minor axis (along y-axis) of ellipse is
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
The line x + 6y = 8 touches the ellipse and the equation of tangent of ellipse is
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE ...(1)
and JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE...(2)
Comparing Eqs. (1) and (2), we get
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
The eccentricity of ellipse is
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.10. If one end of a focal chord AB of the parabola y2 = 8x is at JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE then the equation of the tangent to it at B is    (2020)
(1) 2x + y - 24 = 0
(2) x - 2y + 8 = 0
(3) x + 2y + 8 = 0
(4) 2x - y - 24 = 0
Ans.
(2)
Solution. Let the coordinates of point A be (at2, 2at) and y2 = 8x ⇒ a = 2.
Therefore, JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Now, t1.t2 = -1 ⇒ t2 = 2
Therefore, coordinates of other point of focal chord is (8,8). Hence, the equation of tangent at point B is
8y = 4(x + 8) ⇒ 2y = x + 8 ⇒ x - 2y + 8 = 0

Q.11. If tangents are drawn to the ellipse x2 + 2y2 - 2 at all points on the ellipse other than its four vertices then the mid points of the tangents intercepted between the coordinate axes lie on the curve:    (2019)
(1) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
(2)JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
(3) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
(4) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Ans.
(3)
Solution. Given the equation of ellipse,
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.12. Let the length of the latus rectum of an ellipse with its major axis along x-axis and centre at the origin, be 8. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lie on it?    (2019)
(1) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
(2) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

(3) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
(4) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Ans.
(2)
Solution.
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.13. Let S and S' be the foci of an ellipse and B be any one of the extremities of its minor axis. If ΔS'BS is a right angled triangle with right angle at B and area (ΔS'BS) = 8 sq. units, then the length of a latus rectum of the ellipse is:    (2019)
(1) 4    
(2) 2√2
(3) 4√2    
(4) 2
Ans.
(1)
Solution. 
∵ ΔS'BS is right angled triangle, then
(Slope of 55) x (Slope of BS') = - 1
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
b2 = a2e2   ...(1)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
b= 8   ...(2)
From eqn (1)
a2e2 = 8
Also, JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Hence, required length of latus rectum JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
= 4 units

Q.14. If the tangents on the ellipse 4x2 +y2 = 8 at the points (1,2) and (a, b) are perpendicular to each other, then a2 is equal to:    (2019)
(1) 128/17
(2) 64/17
(3) 4/17
(4) 2/17
Ans.
(4)
Solution.
Since (a,b) touches the given ellipse 4x2 + y2 = 8
∴ 4a2 + b2 = 8   ...(1)
Equation of tangent on the ellipse at the point A (1,2) is:
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
But, also equation of tangent at P(a, b) is:
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Since, tangents are perpendicular to each other.
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
⇒ b = 8a   ...(2)
From (1) & (2) we get:
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.15. In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE then the length of its latus rectum is:    (2019)
(1) 10    
(2) 5    
(3) 8    
(4) 6
Ans.
(2)
Solution.
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.16. If the line x - 2y = 12 is tangent to the ellipse JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE at the point JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE , then the length of the latus rectum of the ellipse is:    (2019)
(1) 9    
(2) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE 
(3) 5    
(4)JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Ans.
(1)
Solution.
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.17. The tangent and normal to the ellipse 3x2 + 5y2 = 32 at the point P(2,2) meet the x-axis at Q and R, respectively. Then the area (in sq. units) of the triangle PQR is:    (2019)
(1) 34/15
(2) 14/3
(3) 16/3
(4) 68/15
Ans.
(4)
Solution.
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Tangent on the ellinse at P is
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.18. If the normal to the ellipse 3x2 + 4y= 12 at a point P on it is parallel to the line, 2x + y = 4 and the tangent to the ellipse at P passes through Q (4,4), then PQ is equal to:    (2019)
(1) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
(2) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
(3) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
(4) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Ans.
(1)
Solution. Slope of tangent on the line 2x +y = 4 at point P is 1/2
Given ellipse is,
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.19. An ellipse, with foci at (0, 2) and (0, -2) and minor axis of length 4, passes through which of the following points ?    (2019)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Ans.
(1)
Solution. Let the equation of ellipse:
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Given that length of minor axis is 4 i.e. a = 4. Also given be = 2
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.20. Axis of a parabola lies along x-axis. If its vertex and focus are at distance 2 and 4 respectively from the origin, on the positive x-axis then which of the following points does not lie on it?    (2019)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Ans.
(2)
Solution. Since, vertex and focus of given parabola is (2, 0) and (4, 0) respectively
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Then, equation of parabola is
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Hence, the point (8, 6) does not lie on given parabola.

Q.21. If θ denotes the acute angle between the curves, y = 10 - x2 and y = 2 + x2 at a point of their intersection, then |tan θ| is equal to:    (2019)
(1) 4/9
(2) 8/15
(3) 7/17
(4) 8/17
Ans. 
(2)
Solution. Since, the equation of curves are
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Differentiate equation (2) with respect to x
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.22. Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Let X denote the random variable of number of aces obtained in the two drawn cards. Then P(X = 1) + P(X = 2) equals:    (2019)
(1) 49/169   
(2) 52/169
(3) 24/169    
(4) 25/169
Ans.
(4)
Solution.
X = number of aces drawn
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.23. Let A (4, -4) and B (9, 6) be points on the parabola, y2 = 4x. Let C be chosen on the arc AOB of the parabola, where O is the origin, such that the area of ΔACB is maximum. Then, the area (in sq. units) of ΔACB, is:    (2019)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
(3) 32
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Ans.
(1)
Solution.
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Let the coordinates of C is (t2, 2t).
Since, area of ΔACB
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.24. If the parabolas y2 = 4b(x - c) and y2 = 8ax have a common normal, then which one of the following is a valid choice for the ordered triad (a, b, c)?    (2019)
(1) (1/2,2,3)    
(2) (1,1,3)
(3) (1/2,2,0)
(4) (1,1,0)
Ans.
(1,2,3,4)
Solution.
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
and normal to y2 = 4b(x- c) with slope m is
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Since, both parabolas have a common normal.
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
or (X-axis is common normal always)

Since, x-axis is a common normal. Hence all the options are correct for m = 0.

Q.25. The length of the chord of the parabola x2 = 4y having equation JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE    (2019)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Ans. (4)
Solution. Let intersection points be P(x1,y1) and Q(x2,y2)
The given equations
x2 =4y   ...(1)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE   ...(2)
Use eqn (1) in eqn (2)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Since, points P and Q both satisfy the equations (2), then
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.26. If the area of the triangle whose one vertex is at the vertex of the parabola, y2 + 4(x - a2) = 0 and the other two vertices are the points of intersection of the parabola and y-axis, is 250 sq. units, then a value of ‘a’ is:    (2019)
(1) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE    
(2) 5(21/3)
(3) (10)2/3    
(4) 5
Ans.
(4)
Solution. y2 = -4(x - a2)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.27. The equation of a tangent to the parabola, x2 = 8y, which makes an angle θ with the positive direction of x-axis, is:    (2019)
(1) y = x tanθ + 2 cotθ
(2) y = x tanθ - 2 cotθ
(3) x = y cotθ + 2 tanθ
(4) x = y cotθ - 2 tanθ
Ans.
(3)
Solution. x2 = 8y
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Then, equation of tangent at P
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.28. The tangent to the parabola y2 = 4x at the point where it intersects the circle x2 + y2  = 5 in the first quadrant, passes through the point:    (2019)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Ans.
(3)
Solution. To find intersection point of x2 + y2 = 5 and y2 = 4x,
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.29.If one end of a focal chord of the parabola, y2 = 16x is at (1,4), then the length of this focal chord is:    (2019)
(1) 25    
(2) 22    
(3) 24    
(4) 20
Ans.
(1)
Solution.
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
One end of focal of the parabola is at (1,4)
∵ y - coordinate of focal chord is 2at
∴ 2 at = 4
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Hence, the required length of focal chord
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.30. The area (in sq. units) of the smaller of the two circles that touch the parabola, y2 = 4x at the point (1,2) and the x - axis is:    (2019)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Ans. 
(4)
Solution. The circle and parabola will have common tangent at P (1, 2).
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
So, equation of tangent to parabola is,
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Let equation of circle (by family of circles) is
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.31. If the tangent to the parabola y2 = x at a point (α, β), (β > 0) is also a tangent to the ellipse, x2 + 2y2 = 1, then a is equal to:    (2019)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Ans.
(4)
Solution. Let tangent to parabola at point JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.32. If the line ax + y = c, touches both the curves x2 + y2 = 1 and JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE, then |c| is equal to:    (2019)
(1) 2
(2) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
(3) 1/2
(4) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Ans. 
(4)
Solution.
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.33. Let P be the point of intersection of the common tangents to the parabola y2 = 12x and hyperbola 8x- y2 = 8. If S and S' denote the foci of the hyperbola where S lies on the positive x-axis, then P divides SS' in a ratio:    (2019)
(1) 13 : 11    
(2) 14 : 13
(3) 5 : 4   
(4) 2 : 1
Ans.
(3)
Solution. Equation of tangent to JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Equation of tangent to
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Q.34. JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEEIf the eccentricity of the hyperbola JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEEis greater than 2, then the length of its latus rectum lies in the interval:    (2019)
(1) (3,∞)    
(2) (3/2,2]
(3) (2,3]    
(4) (1,3/2]
Ans. 
(1)
Solution.
∵ a2 = cos2θ, b2 = sin2θ
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Hence, length of latus rectum lies in the interval (3, ∞)

Q.35. A hyperbola has its centre at the origin, passes through the point (4, 2) and has transverse axis of length 4 along the x - axis. Then the eccentricity of the hyperbola is:    (2019)
(1) 3/2
(2) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
(3) 2
(4) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Ans. 
(4)
Solution.
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Consider equation of hyperbola
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
∵ (4, 2) lies on hyperbola
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.36. The equation of a tangent to the hyperbola 4x2 - 5y2 = 20 parallel to the line x - y = 2 is:    (2019)
(1) x - y + 1 = 0    
(2) x - y + 7 = 0
(3) x - y + 9 = 0    
(4) x - y - 3 = 0
Ans. 
(1)
Solution. Given, the equation of line,
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
The equation of tangent to the hyperbola is,
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.37. Let
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEEJEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE    (2019)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Ans. (2)
Solution. Since, JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE then there are two cases, when r > 1
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Then,
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Then,
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.38. Equation of a common tangent to the parabola y2 = 4x and the hyperbola xy = 2 is:    (2019)
(1) x + y + 1 =0    
(2) x - 2y + 4 = 0
(3) x + 2y + 4 =0    
(4) 4x + 2y + 1=0
Ans.
(3)
Solution. Equation of a tangent to parabola y2 = 4x is:
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
This line is a tangent to xy = 2
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
∵ Tangent is common for parabola and hyperbola.
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.39. If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the eccentricity of the hyperbola is:    (2019)
(1) 13/12
(2) 2
(3) 13/6
(4) 13/8
Ans.
(1)
Solution.
∴ Conjugate axis = 5
∴ 2b = 5
Distance between foci =13
2ae = 13
Then, b2 = a2 (e2 - 1)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.40. If the vertices of a hyperbola be at (-2, 0) and (2, 0) and one of its foci be at (-3, 0), then which one of the following points does not lie on this hyperbola?    (2019)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Ans. 
(4)
Solution. Let the points are,
 JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
⇒ Centre of hyperbola is 0(0, 0)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
∵ Distance between the centre and foci is ae.
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.41. If the eccentricity of the standard hyperbola passing through the point (4, 6) is 2, then the equation of the tangent to the hyperbola at (4, 6) is:    (2019)
(1) x - 2y + 8 = 0    
(2) 2x - 3y + 10 = 0
(3) 2x - y - 2 = 0    
(4) 3x - 2y = 0
Ans. 
(3)
Solution. Let equation of hyperbola be
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE    ...(i)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
On solving (i) and (ii), we get
a2 = 4, b2 = 12
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Now equation of tangent to the hyperbola at (4, 6) is
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.42. If the line y = mx + JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE is normal to the hyperbola JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE then a value of m is:    (2019)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Ans.
(3)
Solution. Since, 1x + my + n = 0 is a normal to JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE 

JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.43. If a directrix of a hyperbola centered at the origin and passing through the point JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE is JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE and its eccentricity is e, then:    (2019)
(1) 4e- 24e2 + 27 = 0    
(2) 4e- 12e- 27 = 0
(3) 4e4 - 24e2 + 35 = 0    
(4) 4e4 + 8e- 35 = 0
Ans.
(3)
Solution.
∵ directrix of a hyperbola is,
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.44. If 5x + 9 = 0 is the directrix of the hyperbola 16x2 - 9y= 144, then its corresponding focus is:    (2019)
(1) (5,0)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
(4) (-5, 0)
Ans. 
(4)
Solution. 
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.45. If the curve y2 = 6x, 9x2 + by2 = 16 intersect each other at right angles, then the value of b is:    (2018)
(1) 6
(2) 7/2
(3) 4
(4) 9/2
Ans.
(4)
Solution. 
Let the point of intersection be (x1, y1) finding slope of both the curves at point of intersection
for y2 = 6x, 9x2 + by2 = 16
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.46. Tangent and normal are drawn at P(16, 16) on the parabola y2 = 16x, which intersect the axis of the parabola at A and B, respectively. If C is the centre of the circle through the points P, A and B and ∠CPB = θ, then a value of tanq is:    (2018)
(1) 1/2
(2) 2
(3) 3
(4) 4/3
Ans.
(2)
Solution. The equation of tangent at P
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
The normal is y = y – 16 = -2(x – 16)
B = (24, 0)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
AB is the diameter
Centre of the circle C = (4, 0)
lope of PB = -2 = m
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.47. Tangents are drawn to the hyperbola 4x2 -y2= 36 at the points P and Q. If these tangents intersect at the point T(0, 3) then the area (in sq. units) of ΔPTQ is:    (2018)
(1) 45√5
(2) 54√3
(3) 60√3
(4) 36√5
Ans.
(1)
Solution. Equation of PQ,
4x (0) - 3y = 36
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.48. If β is one of the angles between the normals to the ellipse, x2 + 3y2 = 9 at the points (3 cosθ, √3 sinθ) and (−3 sin θ√3 cosθ); θ ∈ (0, π/2); then 2cotβ/sin2θ is equal to:    (2018)
(1) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
(2) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
(3) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
(4) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Ans.
(1)
Solution.
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Angle between normal is β
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.49. If the tangents drawn to the hyperbola 4y2 = x2 + 1 intersect the co-ordinate axes at the distinct points A  and B, then the locus of the mid-point of AB is:    (2018)
(1) 4x2 – y2 + 16x2y2 = 0
(2) x2 – 4y2 – 16x2y2 = 0
(3) 4x2 – y2 + 16x2y2 = 0
(4) 4x2 – y2 – 16x2y2 = 0
Ans.
(2)
Solution. 
Let tangent drawn at point (x, y) to the hyperbola 4y2 = x2 + 1 is : 4yy, = xx1 + 1
This tangent intersect co-ordinate axes at A and B respectively then JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Let mid point is M (h,k) then of AB
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Since point P(x1, y1) lies on the hyperbola so
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
from (i) & (ii)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
locus of M

Q.50. Two parabolas with a common vertex and with axes along x-axis and y-axis, respectively, intersect each other in the first quadrant. If the length of the latus rectum of each parabola is 3, then the equation of  the common tangent to the two parabolas is:    (2018)
(1) 8(2x + y) + 3 = 0
(2) 3(x + y) + 4 = 0
(3) 4(x + y) + 3 = 0
(4) x + 2y + 3 = 0
Ans.
(3)
Solution. Equation two parabola are y2 = 3x and x2 = 3y
Let equation of tangent to y2 = 3x is y = mx + 3/4m
is also tangent to x2 = 3y
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
⇒ 4mx2 – 12mx – 9 = 0 have equal roots
⇒ D = 0
⇒ 144 m4 = 4(4m) (–9)
⇒ m4 + m = 0 ⇒ m = – 1
Hence common tangent is JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
4(x + y) + 3 = 0

Q.51. If the length of the latus rectum of an ellipse is 4 units and the distance between a focus and its nearest vertex on the major axis is 3/2 units, then its eccentricity is:    (2018)
(1) 2/3
(2) 1/2
(3) 1/9
(4) 1/3
Ans.
(4)
Solution.
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.52. The locus of the point of intersection of the lines, √2 x – y + 4√2 k = 0 and √2 kx + ky – 4 √2 = 0 (k is any non-zero real parameter), is    (2018)
(1) an ellipse whose eccentricity is 1/√3
(2) a hyperbola whose eccentricity is √3
(3) a hyperbola with length of its transverse axis 8√2
(4) an ellipse with length of its major axis 8√2
Ans.
(3)
Solution.
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.53. Let P be a point on the parabola, x2 = 4y. If the distance of P from the centre of the circle, x2 + y2 + 6x + 8 = 0 is minimum, then the equation of the tangent to the parabola at P, is:    (2018)
(1) x + y + 1 = 0
(2) x + 4y – 2 = 0
(3) x + 2y = 0
(4) x – y + 3 = 0
Ans. 
(1)
Solution. Let P(2t, t2) be any point on the parabola.
Centre of the given circle C = (-g,-f) = (-3,0)
For PC to be minimum, it must be the normal to the parabola at P.
Slope of line PC JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Also, slope of tangent to parabola at JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
∴ Slope of normal JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
∴ Real roots of above equation is t= -1
Coordinate of P = (2t, t2) = (-2,1)
Slope of tangent to parabola at P = t = -1
Therefore, equation of tangent is:
(y-1) = (-1)(x+2)
⇒ x + y + 1 = 0

Q.54. The normal to the curve y (x - 2)(x - 3) = x + 6 at the point where the curve intersects the y-axis passes through the point    (2017)
(1) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
(2) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
(3) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
(4) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Ans.
(3)
Solution. y (x - 2)( x - 3) = x + 6
At y-axis, x = 0, y = 1
Now, on differentiation.
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Now slope of normal = –1
Equation of normal  y – 1 = –1(x – 0)
y + x – 1 = 0 ... (i)
Line (i) passes through (1/2,1/2)

Q.55. A hyperbola passes through the point P(√2,√3) and has foci at (±2, 0). Then the tangent to this hyperbola at P also passes through the point    (2017)
(1) (-√2,-√3)
(2) (3√2, 2√3)
(3) (2√2, 3√3)
(4) (√3,-√2)
Ans.
(3)
Solution.
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
a2 +b2= 4
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
⇒ b2 = 3

∴ a2= 1
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
∴ Tangent at JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Clearly it passes through (2√2, 3√3)

Q.56. The radius of a circle, having minimum area, which touches the curve y = 4 – x2 and the lines, y = |x| is    (2017)
(1) 4 (√2 + 1)
(2) 2 (√2 + 1)
(3) 2 (√2 - 1)
(4) 4 (√2 - 1)
Ans.
(4)
Solution.
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
x2 =-(y - 4)
Let a point on the parabola JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Equation of normal at P is
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
It passes through centre of circle, say (0, k)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
(Length of perpendicular from (0, k) to y = x)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Equation of circle is
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
It passes through point P
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
t4 +t2 (8k- 28) + 8k2 - 128k + 256 = 0
For t = ⇒  k2 - 16k+ 32 = 0
k = 8±4√2
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
(14 - 4k)2+ (14 - 4k) (8k - 28) + 8k2 -128k + 256 = 0
2k2 +4k-15 = 0
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
From (iii) & (iv),
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
But from options, r = 4(√2-1)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.57. The eccentricity of an ellipse whose centre is at the origin is 1/2. If one of its directrices is x = – 4, then the equation of the normal to it at JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE is    (2017)
(1) x + 2y = 4
(2) 2y – x = 2
(3) 4x – 2y = 1
(4) 4x + 2y = 7
Ans. 
(3)
Solution.

JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
x = –4
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
-a= - 4 x e
a = 2
Now,  b2 =a2 (1- e2) = 3
Equation to ellipse
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Equation of normal is
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.58. The tangent at the point (2,–2) to the curve x2y2– 2x = 4(1 – y) does not pass through the point:    (2017)
(1) (–2,–7)
(2) (8,5)
(3) (–4,–9)
(4) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Ans.
(1)
Solution. x2y2–2x = 4 – 4y
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.59. The locus of the point of intersection of the straight lines,
tx – 2y – 3t = 0
x – 2ty + 3 = 0 (t ∈ R) , is:    (2017)
(1) a hyperbola with the length of conjugate axis 3
(2) a hyperbola with eccentricity √5
(3) an ellipse with the length of major axis 6
(4) an ellipse with eccentricity JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Ans.
(1)
Solution.  tx – 2y – 3t = 0
x – 2ty + 3 = 0
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.60. If the common tangents to the parabola, x2 = 4y and the circle, x+ y2 = 4 intersect at the point P, then the distance of P from the origin, is:    (2017)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Ans.
(4)
Solution. 
tangent to x2 + y2 = 4
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.61. Consider an ellipse, whose centre is at the origin and its major axis is along the x - axis. if its eccentricity is 3/5 and the distance between its foci is 6, then the area (in sq. units) of the quadrilateral inscribed in the ellipse, with the vertices as the vertices of the ellipse, is:    (2017)
(1) 32
(2) 80
(3) 40
(4) 8
Ans. 
(3)
Solution.
e = 3/5, 2ae = 6, a(5) a = 5
b2 = a2 (1–e2)
b2 = 25 (1–9/25)
b = 4
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
area = 4 (1/2 ab)
= 2ab = 40

Q.62. If y = mx+c is the normal at a point on the parabola y2=8x whose focal distance is 8 units, then |c| is equal to:    (2017)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Ans. 
(2)
Solution. c = – 29m – 9m3 
a = 2
Given (at2 – a)2 + 4a2t  = 64
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.63. The eccentricity of an ellipse having centre at the origin, axes along the co-ordinate axes and passing through the points (4, –1) and (–2,2) is:    (2017)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Ans.
(1)
Solution. e = ?, centre at (0,0)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
16b2 + a2 = a2b   ...(1)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
4b2 + 4a2 = a2b2   ...(2)
From (1) & (2)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.64. Let P be the point on the parabola, y2 = 8x which is at a minimum distance from the centre C of the circle, x2 + (y + 6)2 = 1. Then the equation of the circle, passing through C and having its centre at P is:    (2016) 
(1) x2 + y2 - 4x + 8y + 12 = 0 
(2) x2 + y2 - x + 4y - 12 = 0 
(3) x2 + y2 - JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE+ 2y - 24 = 0 
(4) x2 + y2 - 4x + 9y + 18 = 0 
Ans.
(1)
Normal at P(at2, 2at) is y + tx = 2at + at3 Given it passes (0, -6) 
 -6 = 2at + at(a = 2) 
-6 = 4t + 2t3 
t3 + 2t + 3 = 0 
t = -1 
so, P (a, -2a) = (2, -4) . [a = 1) 
radius of circle = CP = JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Circle is (x - 2)2 + (y + 4)2 = JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
x2 + y2 - 4x + 8y + 12 = 0 

Q.65. The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half the distance between its foci, is:    (2016)
(1) 4/3
(2) 4/√3
(3) 2/√3
(4) √3
Ans.
(3)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Squaring eqn. (2), we get
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEEand we know that JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.66. If the tangent at a point on the ellipse JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE meets the coordinate axes at A and B, and O is the origin, then the minimum area (in sq. units) of the triangle OAB is:    (2016)
(1) 9
(2) 9/2
(3) 9√3
(4) 3√3
Ans.
(1)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.67. The minimum distance of a point on the curve y = x2 - 4 from the origin is:    (2016)
(1) √15/2
(2) √19/2
(3) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
(4) JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
Ans.
(1)
Let point at minimum distance from O is
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.68.  Let a and b respectively be the semi-transverse and semi-conjugate axes of a hyperbola whose eccentricity satisfies the equation 9e2 - 18e + 5 = 0. If S(5, 0) is a focus and 5x = 9 is the corresponding directrix of hyperbola, then a2 - b2 is equal to    (2016)
(1) -7
(2) -5
(3) 5
(4) 7
Ans.
(1)
9e- 18e + 5 = 0
⇒ e = 5/3
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE   (i)
Also distance between foci and directrix is
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.69. If the tangent at a point P, with parameter t, on the curve x = 4t2 + 3, y = 8t3 - 1, t ∈ R, meets the curve again at a point Q, then the coordinates of Q are:    (2016)
(1) (t2 + 3, - t3 - 1)
(2) (t2 + 3, t3 - 1)
(3) (16t+ 3, - 64t3 - 1)
(4) (4t2 + 3, - 8t3 - 1)
Ans.
(1)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
∴ Q(t2 + 3, - t3 - 1)

Q.70. P and Q are two distinct points on the parabola, y2  = 4x, with parameters t and t1 respectively. If the normal at p passes through Q, then the minimum value of t12 is    (2016)
(1) 4
(2) 6
(3) 8
(4) 2
Ans.
(3)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.71. A hyperbola whose transverse axis is along the major axis of the conic, JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE and has vertices at the foci of this conic. If the eccentricity of the hyperbola is 3/2, then which of the following points does NOT lie on it?    (2016)
(1) √5, 2√2
(2) 5, 2√3
(3) 0, 2
(4) √10, 2√3
Ans.
(2)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE

Q.72. Let C be a curve given by y(x) = JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEEIf P is a point on C, such that the tangent at P has slope 2/3,  then a point through which the normal at P passes, is    (2016)
(1) 3, -4
(2) 1, 7
(3) 4, -3
(4) 2, 3
Ans.
(2)
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Conic Sections Notes | Study Mathematics For JEE - JEE
2y - 8 = - 3x + 9
3x + 2y = 17
clearly it is passes through (1, 7)

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