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Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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1. The line 2x + y = 1 is tangent to the hyperbola  Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 If this line passes through the point of intersection of the nearest directrix and the x-axis, then the eccentricity of the hyperbola is (2010) 

Ans.  (2)

Sol.  Intersection point of nearest directrix x = Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEand x-axis

is Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

As 2x +y=1 passes throughInteger Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Also y = –2x+1 is a tangent to Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ 4a2– a2e2 – 1=1 ⇒ Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE 

⇒ 4e–e4 +e= 4 ⇒ e4 – 5e4 + 4=0

⇒ e2 = 4 ase>1 for hyperbola.  ⇒ e= 2

 

 

2. Consider the parabola y2 = 8x . Let Δ1 be the area of the triangle formed by the end points of its latus rectum and the pointInteger Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEon the parabola and Δ2 be the area of the triangle formed by drawing tangents at P and at the end points of the latus rectum. Then Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is     (2011) 

Ans. (2)

Sol. Δ1 =  Area of ΔPLL'  Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Equation of AB,  y = 2x + 1 Equation of AC, y = x + 2 Equation of BC,  – y = x + 2 Solving above equations we get A (1, 3), B (–1, –1), C (–2, 0)

Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

 

 

3. Let S be the focus of the parabola y2  = 8x and let PQ be the common chord of the circle x2 + y2 – 2x – 4y =  0 and the given parabola. The area of the triangle PQS is     (2012)

Ans.  (4)

Sol.  We observe both parabola y2 = 8x and circle x2 + y2 – 2x – 4y = 0 pass through origin
∴ One end of common chord PQ is origin. Say P(0, 0)
Let Q be the point (2t2, 4t), then it will satisfy the equation of circle.
∴ 4t4 + 16t2 – 4t2 – 16t = 0 ⇒ t4 + 3t2 – 4t = 0  ⇒ t (t+ 3t – 4) = 0
⇒ t (t – 1)(t2 + t – 4) = 0   ⇒ t = 0 or 1
For t = 0,  we get point P, therefore t = 1 gives point  Q as (2, 4).
We also observe here that P(0, 0) and Q(2, 4) are end points of diameter of the given circle and focus of the parabola is the point S(2, 0).

∴ Area of  ΔPQS Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEsq. units

Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

 

 

4. A vertical line passing through the point (h, 0) intersects the
 ellipse Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

at the points P and Q. Let the tangents to the ellipse at P and Q meet at the point R. If Δ(h) = area of the triangle  Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE then

Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE(JEE Adv. 2013)
 (a) g(x) is continuous but not differentiable at a
 (b) g(x) is differentiable on R
 (c) g(x) is continuous but not differentiable at b
 (d) g(x) is continuous and differentiable at either (a) or (b) but not both

Ans. (9)

Sol. Vertical line x = h, meets the ellipse  Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEat

Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEand  Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

By symmetry, tangents at P and Q will meet each other at x-axis.

Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Tangent at P is  Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

which meets x-axis at Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Area of  ΔPQR  = Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

i.e.,Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ Δ(h) is a decreasing function.

Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEInteger Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEInteger Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE= 45 – 36 = 9

 

 

5. If the normals of the parabola y2 = 4x drawn at the end points of its latus rectum are tangents to the circle (x – 3)2 + (y + 2)2 = r2, then the value of ris (JEE Adv. 2015)

Ans. (2)

Sol. End points of latus rectum of y2 = 4x are (1, +2)
Equation of normal to y= 4x at (1, 2) is y – 2 = –1(x – 1)
or x + y –3 = 0
As it is tangent to circle (x – 3)2 + (y + 2)2 = r2

Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

 

 

6. Let the curve C be the mirror image of the parabola y2 = 4x with respect to the line x + y + 4 = 0. If A and B are the points of intersection of C with the line  y = –5, then the distance between A and B is (JEE Adv. 2015)

Ans. (4) 

Sol. Let (t2, 2t) be any point on y2 = 4x. Let (h, k) be the image of (t2, 2t) in the line x + y + 4 = 0. Then

Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ h = –(2t + 4) and k = –(t2 + 4)
For its intersection with, y = –5, we have –(t2 + 4) = –5 ⇒ t = +1
∴ A(–6, –5) and B(–2, –5) ∴ AB = 4.

 

 

7. Suppose that the foci of the ellipse

Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE = 1 are (f1, 0)and (f2, 0) where f1 > 0 and f2 < 0. Let P1 and P2 be two parabolas with a common vertex at (0, 0) and with foci at (f1, 0) and (2f2, 0), respectively. Let T1 be a tangent to P1 which passes through (2f2, 0) and Tbe a tangent to P2 which passes through (f1, 0). If m1 is the slope of T1 and m2 is the slope of T2, then the value of

Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is (JEE Adv. 2015)

Ans. (4)

Sol.  Ellipse :Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ a = 3, b = Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ f1 = 2 and f= –2 P1 : y2 = 8x and P2 : y2 = –16x

T1 : y = m1x  +Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

It passes through (–4, 0),

Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

It passes through (2, 0)

Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

The document Integer Answer Type Questions: Conic Sections | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is a part of the JEE Course Maths 35 Years JEE Main & Advanced Past year Papers.
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