Question

A line is a common tangent to the circle $(x-3)^{2}+y^{2}=9$ and the parabola $y^{2}=4 x$ If the two points of contact $( a , b )$ and $( c , d )$ are distinct and lie in the first quadrant, then $2(a+c)$ is equal to______.

Answer 1

Let coordinate of point $A \left( t ^{2}, 2 t \right) \quad(\because a =1)$
equation of tangent at point $A$
$yt = x + t ^{2}$
$x-t y+t^{2}=0$
centre of circle (3,0)
Now $PD =$ radius
$\left|\frac{3-0+t^{2}}{\sqrt{1+t^{2}}}\right|=3$
$\left(3+t^{2}\right)^{2}=9\left(1+t^{2}\right)$
$9+t^{4}+6 t^{2}=9+9 t^{2}$
$t =0,-\sqrt{3}, \sqrt{3}$
So point $A (3,2 \sqrt{3})$$\Rightarrow a =3, b =2 \sqrt{3}$
(Since it lies in first quadrant)
For point $B$ which is foot of perpendicular from
centre (3,0) to the tangent $x-\sqrt{3} y+3=0$
$\frac{c-3}{1}=\frac{d-0}{-\sqrt{3}}=\frac{-(3-0+3)}{4}$
$\Rightarrow c=\frac{3}{2} d=\frac{3 \sqrt{3}}{2}$
$\Rightarrow 2\left(\frac{3}{2}+3\right)=9$