Previous year Questions (2016-20) - Conic Sections Notes | EduRev

Maths 35 Years JEE Mains & Advance Past year Papers Class 11

JEE : Previous year Questions (2016-20) - Conic Sections Notes | EduRev

The document Previous year Questions (2016-20) - Conic Sections Notes | EduRev is a part of the JEE Course Maths 35 Years JEE Mains & Advance Past year Papers Class 11.
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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
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Now, Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Hence, the length of latus rectum is
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Therefore, the equation of tangent is y = 4x + 1/4. It is also the tangent of parabola x2 = 2by, then
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
The value of b cannot be zero, hence b = −128.

Q.3. If 3x + 4y = 12√2 is a tangent to the ellipse Previous year Questions (2016-20) - Conic Sections Notes | EduRev for some Previous year Questions (2016-20) - Conic Sections Notes | EduRevthen 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 Previous year Questions (2016-20) - Conic Sections Notes | EduRevis
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
The line y = mx + c will be a tangent of ellipse Previous year Questions (2016-20) - Conic Sections Notes | EduRev if
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Now, the foci of ellipse is Previous year Questions (2016-20) - Conic Sections Notes | EduRev
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 Previous year Questions (2016-20) - Conic Sections Notes | EduRev 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
Previous year Questions (2016-20) - Conic Sections Notes | EduRev...(1)
It passes through the point Previous year Questions (2016-20) - Conic Sections Notes | EduRev then
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Now, Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev (since P lies in first quadrant)
The normal also passes through the point(0, β), then
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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,
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Now,
Previous year Questions (2016-20) - Conic Sections Notes | EduRev ...(1)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev ...(2)
From Eqs. (1) and (2), we get
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
The coordinates of vertices are (±a, 0) = (±6, 0) ⇒ a = 6.
The equation of hyperbola is
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Now, hyperbola passes through the point P (10, 16), then
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Therefore, the equation of hyperbola is
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Now, the equation of normal at point P is
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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 Previous year Questions (2016-20) - Conic Sections Notes | EduRev 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.

Previous year Questions (2016-20) - Conic Sections Notes | EduRev


Q.8. If e1 and e2 are the eccentricities of the ellipse Previous year Questions (2016-20) - Conic Sections Notes | EduRev and the hyperbola Previous year Questions (2016-20) - Conic Sections Notes | EduRev 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 Previous year Questions (2016-20) - Conic Sections Notes | EduRevis
Previous year Questions (2016-20) - Conic Sections Notes | EduRev ...(1)
The eccentricity of hyperbola Previous year Questions (2016-20) - Conic Sections Notes | EduRev is
Previous year Questions (2016-20) - Conic Sections Notes | EduRev ...(2)
Point (e1, e2) lies on the ellipse 15x2 + 3y2 = k, then
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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) Previous year Questions (2016-20) - Conic Sections Notes | EduRev
(2) Previous year Questions (2016-20) - Conic Sections Notes | EduRev
(3) Previous year Questions (2016-20) - Conic Sections Notes | EduRev
(4) Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Ans.
(1)
Solution. The length of the minor axis (along y-axis) of ellipse is
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
The line x + 6y = 8 touches the ellipse and the equation of tangent of ellipse is
Previous year Questions (2016-20) - Conic Sections Notes | EduRev ...(1)
and Previous year Questions (2016-20) - Conic Sections Notes | EduRev...(2)
Comparing Eqs. (1) and (2), we get
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
The eccentricity of ellipse is
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

Q.10. If one end of a focal chord AB of the parabola y2 = 8x is at Previous year Questions (2016-20) - Conic Sections Notes | EduRev 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, Previous year Questions (2016-20) - Conic Sections Notes | EduRev
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) Previous year Questions (2016-20) - Conic Sections Notes | EduRev
(2)Previous year Questions (2016-20) - Conic Sections Notes | EduRev
(3) Previous year Questions (2016-20) - Conic Sections Notes | EduRev
(4) Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Ans.
(3)
Solution. Given the equation of ellipse,
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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) Previous year Questions (2016-20) - Conic Sections Notes | EduRev
(2) Previous year Questions (2016-20) - Conic Sections Notes | EduRev

(3) Previous year Questions (2016-20) - Conic Sections Notes | EduRev
(4) Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Ans.
(2)
Solution.
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
b2 = a2e2   ...(1)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
b= 8   ...(2)
From eqn (1)
a2e2 = 8
Also, Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Hence, required length of latus rectum Previous year Questions (2016-20) - Conic Sections Notes | EduRev
= 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:
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
But, also equation of tangent at P(a, b) is:
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Since, tangents are perpendicular to each other.
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
⇒ b = 8a   ...(2)
From (1) & (2) we get:
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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 Previous year Questions (2016-20) - Conic Sections Notes | EduRev then the length of its latus rectum is:    (2019)
(1) 10    
(2) 5    
(3) 8    
(4) 6
Ans.
(2)
Solution.
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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

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.
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Tangent on the ellinse at P is
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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) Previous year Questions (2016-20) - Conic Sections Notes | EduRev
(2) Previous year Questions (2016-20) - Conic Sections Notes | EduRev
(3) Previous year Questions (2016-20) - Conic Sections Notes | EduRev
(4) Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Ans.
(1)
Solution. Slope of tangent on the line 2x +y = 4 at point P is 1/2
Given ellipse is,
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Ans.
(1)
Solution. Let the equation of ellipse:
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Given that length of minor axis is 4 i.e. a = 4. Also given be = 2
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Ans.
(2)
Solution. Since, vertex and focus of given parabola is (2, 0) and (4, 0) respectively
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Then, equation of parabola is
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
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
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Differentiate equation (2) with respect to x
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
(3) 32
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Ans.
(1)
Solution.
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Let the coordinates of C is (t2, 2t).
Since, area of ΔACB
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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.
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
and normal to y2 = 4b(x- c) with slope m is
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Since, both parabolas have a common normal.
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
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 Previous year Questions (2016-20) - Conic Sections Notes | EduRev    (2019)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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

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) Previous year Questions (2016-20) - Conic Sections Notes | EduRev    
(2) 5(21/3)
(3) (10)2/3    
(4) 5
Ans.
(4)
Solution. y2 = -4(x - a2)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Then, equation of tangent at P
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Ans.
(3)
Solution. To find intersection point of x2 + y2 = 5 and y2 = 4x,
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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.
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
One end of focal of the parabola is at (1,4)
∵ y - coordinate of focal chord is 2at
∴ 2 at = 4
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Hence, the required length of focal chord
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Ans. 
(4)
Solution. The circle and parabola will have common tangent at P (1, 2).
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
So, equation of tangent to parabola is,
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Let equation of circle (by family of circles) is
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Ans.
(4)
Solution. Let tangent to parabola at point Previous year Questions (2016-20) - Conic Sections Notes | EduRev Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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

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 Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Equation of tangent to
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Q.34. Previous year Questions (2016-20) - Conic Sections Notes | EduRevIf the eccentricity of the hyperbola Previous year Questions (2016-20) - Conic Sections Notes | EduRevis 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
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
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) Previous year Questions (2016-20) - Conic Sections Notes | EduRev
(3) 2
(4) Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Ans. 
(4)
Solution.
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Consider equation of hyperbola
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
∵ (4, 2) lies on hyperbola
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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,
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
The equation of tangent to the hyperbola is,
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

Q.37. Let
Previous year Questions (2016-20) - Conic Sections Notes | EduRevPrevious year Questions (2016-20) - Conic Sections Notes | EduRev    (2019)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Ans. (2)
Solution. Since, Previous year Questions (2016-20) - Conic Sections Notes | EduRev then there are two cases, when r > 1
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Then,
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Then,
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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:
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
This line is a tangent to xy = 2
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
∵ Tangent is common for parabola and hyperbola.
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Ans. 
(4)
Solution. Let the points are,
 Previous year Questions (2016-20) - Conic Sections Notes | EduRev
⇒ Centre of hyperbola is 0(0, 0)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
∵ Distance between the centre and foci is ae.
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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
Previous year Questions (2016-20) - Conic Sections Notes | EduRev    ...(i)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
On solving (i) and (ii), we get
a2 = 4, b2 = 12
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Now equation of tangent to the hyperbola at (4, 6) is
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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

Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

Q.43. If a directrix of a hyperbola centered at the origin and passing through the point Previous year Questions (2016-20) - Conic Sections Notes | EduRev is Previous year Questions (2016-20) - Conic Sections Notes | EduRev 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,
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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

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
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
The normal is y = y – 16 = -2(x – 16)
B = (24, 0)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
AB is the diameter
Centre of the circle C = (4, 0)
lope of PB = -2 = m
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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) Previous year Questions (2016-20) - Conic Sections Notes | EduRev
(2) Previous year Questions (2016-20) - Conic Sections Notes | EduRev
(3) Previous year Questions (2016-20) - Conic Sections Notes | EduRev
(4) Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Ans.
(1)
Solution.
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Angle between normal is β
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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 Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Let mid point is M (h,k) then of AB
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Since point P(x1, y1) lies on the hyperbola so
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
from (i) & (ii)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
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
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
⇒ 4mx2 – 12mx – 9 = 0 have equal roots
⇒ D = 0
⇒ 144 m4 = 4(4m) (–9)
⇒ m4 + m = 0 ⇒ m = – 1
Hence common tangent is Previous year Questions (2016-20) - Conic Sections Notes | EduRev
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.
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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.
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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 Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Also, slope of tangent to parabola at Previous year Questions (2016-20) - Conic Sections Notes | EduRev
∴ Slope of normal Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
∴ 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) Previous year Questions (2016-20) - Conic Sections Notes | EduRev
(2) Previous year Questions (2016-20) - Conic Sections Notes | EduRev
(3) Previous year Questions (2016-20) - Conic Sections Notes | EduRev
(4) Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Ans.
(3)
Solution. y (x - 2)( x - 3) = x + 6
At y-axis, x = 0, y = 1
Now, on differentiation.
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
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.
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
a2 +b2= 4
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
⇒ b2 = 3

∴ a2= 1
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
∴ Tangent at Previous year Questions (2016-20) - Conic Sections Notes | EduRev
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.
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
x2 =-(y - 4)
Let a point on the parabola Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Equation of normal at P is
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
It passes through centre of circle, say (0, k)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
(Length of perpendicular from (0, k) to y = x)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Equation of circle is
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
It passes through point P
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
t4 +t2 (8k- 28) + 8k2 - 128k + 256 = 0
For t = ⇒  k2 - 16k+ 32 = 0
k = 8±4√2
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
(14 - 4k)2+ (14 - 4k) (8k - 28) + 8k2 -128k + 256 = 0
2k2 +4k-15 = 0
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
From (iii) & (iv),
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
But from options, r = 4(√2-1)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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 Previous year Questions (2016-20) - Conic Sections Notes | EduRev is    (2017)
(1) x + 2y = 4
(2) 2y – x = 2
(3) 4x – 2y = 1
(4) 4x + 2y = 7
Ans. 
(3)
Solution.

Previous year Questions (2016-20) - Conic Sections Notes | EduRev
x = –4
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
-a= - 4 x e
a = 2
Now,  b2 =a2 (1- e2) = 3
Equation to ellipse
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Equation of normal is
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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) Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Ans.
(1)
Solution. x2y2–2x = 4 – 4y
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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 Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Ans.
(1)
Solution.  tx – 2y – 3t = 0
x – 2ty + 3 = 0
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Ans.
(4)
Solution. 
tangent to x2 + y2 = 4
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
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)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Ans. 
(2)
Solution. c = – 29m – 9m3 
a = 2
Given (at2 – a)2 + 4a2t  = 64
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Ans.
(1)
Solution. e = ?, centre at (0,0)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
16b2 + a2 = a2b   ...(1)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
4b2 + 4a2 = a2b2   ...(2)
From (1) & (2)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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 - Previous year Questions (2016-20) - Conic Sections Notes | EduRev+ 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 = Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Circle is (x - 2)2 + (y + 4)2 = Previous year Questions (2016-20) - Conic Sections Notes | EduRev
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)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Squaring eqn. (2), we get
Previous year Questions (2016-20) - Conic Sections Notes | EduRevand we know that Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

Q.66. If the tangent at a point on the ellipse Previous year Questions (2016-20) - Conic Sections Notes | EduRev 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)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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) Previous year Questions (2016-20) - Conic Sections Notes | EduRev
(4) Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Ans.
(1)
Let point at minimum distance from O is
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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
Previous year Questions (2016-20) - Conic Sections Notes | EduRev   (i)
Also distance between foci and directrix is
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
∴ 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)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

Q.71. A hyperbola whose transverse axis is along the major axis of the conic, Previous year Questions (2016-20) - Conic Sections Notes | EduRev 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)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev

Q.72. Let C be a curve given by y(x) = Previous year Questions (2016-20) - Conic Sections Notes | EduRevIf 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)
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
Previous year Questions (2016-20) - Conic Sections Notes | EduRev
2y - 8 = - 3x + 9
3x + 2y = 17
clearly it is passes through (1, 7)

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past year papers

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mock tests for examination

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video lectures

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Viva Questions

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practice quizzes

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Sample Paper

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Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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Semester Notes

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Previous year Questions (2016-20) - Conic Sections Notes | EduRev

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Exam

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Free

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Summary

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