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 $(a, b)$ with respect to the circle and $(c, d)$ with respect to the parabola are distinct and lie in the first quadrant, then $b + 2 d =$ _____________.

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$5 \sqrt{3}$

$3 \sqrt{3}$

$2 \sqrt{3}$

$6 \sqrt{3}$

answer is D.

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Detailed Solution

Equation of the tangent to the parabola is $x - yt + t^{2} = 0$
It is also tangent to the circle
$\begin{array}{l} \Rightarrow r = d \Rightarrow 3 = \frac{|3 - 0 + t^{2}|}{\sqrt{1 + t^{2}}} \\ \Rightarrow 3 \sqrt{1 + t^{2}} = t^{2} + 3 \\ \Rightarrow 9 (1 + t^{2}) = t^{4} + 9 + 6 t^{2} \\ \Rightarrow t^{4} - 3 t^{2} = 0 \Rightarrow t = 0, \pm \sqrt{3} \\ t = \sqrt{3} \Rightarrow (a, b) = (3, 2 \sqrt{3}) \\ (t^{2}, 2 t) \end{array}$
$∴$ Tangent is $x - \sqrt{3} y + 3 = 0$
$\begin{aligned} (c, d) = (\frac{3}{2}, \frac{3 \sqrt{3}}{2}) \\ ∴ b + 2 d = 5 \sqrt{3} . \end{aligned}$