Equation of the common tangent to the parabola $y^{2} = 24 x$ and the circle $x^{2} + y^{2} = 18$ is

x+y+6=0

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x+y+4=0

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2x+y-9=0

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x+2y-8=0

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Solution

The correct option is A
x+y+6=0

$y^{2} = 24 x x^{2} + y^{2} = 18$
$y = m x + \frac{6}{m} 18 = \frac{6^{2}}{m^{4} + m^{2}}$
$\Rightarrow m y = m^{2} x + 6 \Rightarrow m^{4} + m^{2} = 2$
$\Rightarrow m^{2} x - m y + 6 = 0 \Rightarrow m = \pm 1$
$y = \pm (x + 6)$
x+y+6=0
x-y+6=0

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