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