Question

A line is a common tangent to the circle $(x-3{)}^2+y^2=9$ and the parabola $y^2=4x$ 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

Answer: 9

Let coordinate of point $A\left(t^2,2t\right)(\because a=1)$
equation of tangent at point $A$
$yt=x+t^2$
$x-ty+t^2=0$
centre of circle (3,0)
Now $PD=$ radius
$\left|\frac{3-0+t^2}{\sqrt{1+t^2}}\right|=3$
${\left(3+t^2\right)}^2=9\left(1+t^2\right)$
$9+t^4+6t^2=9+9t^2$
$t=0,-\sqrt{3},\sqrt{3}$
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 $x-\sqrt{3}y+3=0$
$\frac{c-3}{1}=\frac{d-0}{-\sqrt{3}}=\frac{-(3-0+3)}{4}$
$\Rightarrow c=\frac{3}{2}d=\frac{3\sqrt{3}}{2}$
$\Rightarrow 2\left(\frac{3}{2}+3\right)=9$