Question

Chords of the parabola $y^2=4 x$ 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 $y^2=4 x$, so equation of chord of contact of $(h, k)$ is $k y=2(x+h)$ .....(2)
As, (1) and (2) are identical, so
$ \frac{\sec \theta}{2}=\frac{\tan \theta}{ k }=\frac{-1}{2 h } \text { (on comparing) } $

$\Rightarrow \sec \theta=\frac{-1}{ h } \text { and } \tan \theta=\frac{ k }{2 h } $
$\text { As, } \sec ^2 \theta-\tan ^2 \theta=1 $
$\Rightarrow \frac{1}{ h ^2}-\frac{ k ^2}{4 h ^2}=1 \Rightarrow 4- k ^2=4 h ^2 $
$\therefore \text { Locus of } P ( h , k ) \text { is } $
$\text { E : } 4 x ^2+ y ^2=4 \text {, where is an ellipse or } \frac{ x ^2}{1}+\frac{ y ^2}{4}=1$ ....(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.