Question

Chords of the parabola $y^2=4x$ are tangent to the hyperbola $x^2-y^2=1$. The locus of the point of intersection of tangents of parabola drawn at extremities of such chords is an ellipse $E$.
The equation of common tangent(s) to hyperbola $x^2-y^2=1$ and ellipse $E$ can be

• A. $x=1$
• B. $x=-1$
• C. $y=1$
• D. $y=-1$

Answer 1

Answer: (A), (B)

Any tangent to given parabola is
$x\cdot \sec \theta -y\cdot \tan \theta =1$ .....(1)
Let $P(h,k)$ be the point of intersection of tangents to the parabola $y^2=4x$, so equation of chord of contact of $(h,k)$ is $ky=2(x+h)$ .....(2)
As, (1) and (2) are identical, so
$\frac{\sec \theta }{2}=\frac{\tan \theta }{k}=\frac{-1}{2h}\text{ (on comparing) }$

$\Rightarrow \sec \theta =\frac{-1}{h}\text{ and }\tan \theta =\frac{k}{2h}$
$\text{ As, }{\sec }^2\theta -{\tan }^2\theta =1$
$\Rightarrow \frac{1}{h^2}-\frac{k^2}{4h^2}=1\Rightarrow 4-k^2=4h^2$
$\therefore \text{ Locus of }P(h,k)\text{ is }$
$\text{ E : }4x^2+y^2=4\text{, where is an ellipse or }\frac{x^2}{1}+\frac{y^2}{4}=1$ ....(3)
From above figure, the equation of common tangent to hyperbola $x^2-y^2=1$ and ellipse $E$ can be $x=-1$ or $x=1.\Rightarrow (A)$ and $(B)$ are correct.