Question

If chord of contact of a point $P$ on the parabola $y^2=4x$ 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

Answer: 5

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