Let a , b and be positive real numbers. Suppose P is an end point of the latus rectum of the parabola y^2=4 x, and suppose the ellipse x^2 a^2 +y^2 b^2 =1 passes through the point P. If the tangents to the parabola and the ellipse at the point P are perpendicular to each other, then the eccentricity of the ellipse is

Let a , b and be positive real numbers. Suppose P is an end point…

MCQ

Let $a , b$ and $\lambda$ be positive real numbers. Suppose $P$ is an end point of the latus rectum of the parabola $y^2=4 \lambda x$, and suppose the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ passes through the point $P$. If the tangents to the parabola and the ellipse at the point $P$ are perpendicular to each other, then the eccentricity of the ellipse is

✓
$\frac{1}{\sqrt{2}}$
B
$\frac{1}{2}$
C
$\frac{1}{3}$
D
$\frac{2}{5}$

✓

Answer

Correct option: A.

$\frac{1}{\sqrt{2}}$

a
$y ^2=4 \lambda x , P (\lambda, 2 \lambda)$

Slope of the tangent to the parabola at point $P$

$\frac{ dy }{ dx }=\frac{4 \lambda}{2 y }=\frac{4 \lambda}{2 x 2 \lambda}=1$

Slope of the tangent to the ellipse at $P$

$\frac{2 x }{ a ^2}+\frac{2 yy ^{\prime}}{ b ^2}=0$

As tangents are perpendicular $y^{\prime}=-1$

$\Rightarrow \frac{2 \lambda}{a^2}-\frac{4 \lambda}{b^2}=0 \Rightarrow \frac{a^2}{b^2}=\frac{1}{2}$

$\Rightarrow e=\sqrt{1-\frac{1}{2}}=\frac{1}{\sqrt{2}}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "MCQ" from 10.2 parabola,ellipse,hyperbola→10.2 parabola,ellipse,hyperbola — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

10.2 parabola,ellipse,hyperbola — MATHS STD 11 Science — Question

Answer

Need a full question paper?

Explore more

Similar questions