Locus of mid points of chords of the parabola $y^{2} = 4 a x$ parallel to the line $l x + m y + n = 0$ is

21 y + a m = 0

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l y + 4 a m = 0

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l y + 3 a m = 0

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l y + 2 a m = 0

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Solution

The correct option is C $l y + 2 a m = 0$
Let the middle points of chord be $(h, k)$
The equation of chord of contact is $S_{1} = S_{11}$
$\Rightarrow k y - 2 a (x + h) = k^{2} - 4 a h$

Now, slope of chord of contact $= \frac{- l}{m} = \frac{2 a}{k}$
$\Rightarrow l k + 2 a m = 0$
Therefore, locus is $l y + 2 a m = 0$

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