Question

If chord of contact of a point $P$ on the parabola $y^2=4 x$ w.r.t. an ellipse $\frac{x^2}{a}+\frac{y^2}{b}=1$ touches the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and focal distance of point $P$ be ' $d$ ', then find $[d]$.
[Note: where $[k]$ denotes greatest integer function less than or equal to $k$.]

Answer 1

Let $P$ be $\left( t ^2, 2 t \right)$
Equation of chord of contact will be $\frac{x \cdot t^2}{a}+\frac{y \cdot 2 t}{b}=1$.....(1)
Equation of tangent on hyperbola is $\frac{x \sec \theta}{a}-\frac{y \tan \theta}{b}=1$....(2)
$\Theta(1) \&(2)$ represent same line
$\therefore t ^2=\sec \theta \text { and } 2 t =-\tan \theta $
$\therefore t ^4-4 t ^2=1 \Rightarrow\left( t ^2-2\right)^2=5 \Rightarrow t ^2=2+\sqrt{5}$
Focal distance, $d = t ^2+1=3+\sqrt{5}$
$\therefore[ d ]=5 $