If a line along a chord of the circle $4 x^{2} + 4 y^{2} + 120 x + 675 = 0,$ passes through the point $(- 30, 0)$ and is tangent to the parabola $y^{2} = 30 x,$ then the length of this chord is

3 \sqrt{5}

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5 \sqrt{3}

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7

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5

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Solution

The correct option is A $3 \sqrt{5}$
Circle $: (x + 15)^{2} + y^{2} = {(\frac{15}{2})}^{2}$

Equation of tangent is
$t y = x + \frac{15}{2} t^{2} \dots (i)$
$(i)$ passes through $(- 30, 0)$
$\Rightarrow 0 = - 30 + \frac{15}{2} t^{2}$
$\Rightarrow t = \pm 2$

$(i) \Rightarrow \pm 2 y = x + 30$
$\Rightarrow x - 2 y + 30 = 0 or x + 2 y + 30 = 0$
Consider $x - 2 y + 30 = 0$
$C S = ∣ ∣ ∣ \frac{- 15 - 0 + 30}{\sqrt{5}} ∣ ∣ ∣ = 3 \sqrt{5}$
$P S = \sqrt{C P^{2} - C S^{2}} = \sqrt{{(\frac{15}{2})}^{2} - (3 \sqrt{5})^{2}} = \frac{3 \sqrt{5}}{2}$
$P Q = 2 P S = 3 \sqrt{5}$

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