Question

Let a circle whose center on the axes touches the parabola $y^2=4x$ at two points such that pair of common tangents of the curves makes an angle of $\frac{\pi }{2}$. If the area of the circle is $k\pi$, then the value of $k$ is

Answer 1

$\because$ Parabola is symmetric about $x-$axis
$\therefore$ Center of the circle will also lie on the $x-$axis and tangents will make an angle of $\pm \frac{\pi }{4}$ with $x-$axis.
Now, slope of the tangent is $m=\pm \tan 45°=\pm 1$
Let the tangents touches the parabola at $(t^2,2t)$
Then ,$m=\frac{1}{t}\Rightarrow t=\pm 1$
So, the coordinates where circle and parabola touches are $(1,2)$ and $(1,-2)$
$\therefore AD=2\sqrt{2}$
$\because △ACD$ is rightangle isosceles triangle
$\therefore AC=2\sqrt{2}$ and $AC$ is radius of the circle
So,area of the circle is $8\pi .$