The locus of the midpoints of the focal chords of the parabola $y^{2} = 4 a x$ is

y^{2} = 2 a (x + a)

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y^{2} = 2 a (x - a)

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y^{2} = 2 a (2 x + a)

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y^{2} = a (2 x - a)

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Solution

The correct option is B $y^{2} = 2 a (x - a)$
Equation of chord with $(x_{1}, y_{1}$ as the midpoint is given by $T = S_{1}$
Equation of chord having $(x_{1}, y_{1})$ as mid point is $y y_{1} - 2 a x = y_{1}^{2} - 2 a x_{1}$
It passes through $(a, 0) \Rightarrow - 2 a^{2} = y_{1}^{2} - 2 a x_{1}$ . Locus is $y^{2} = 2 a (x - a)$

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y^{2} = 4 a x

The locus of the midpoints of the focal chords of the parabola $y^{2} = 4 a x$ is

y^{2} = 4 a x

x^{2} - y^{2} = a^{2}

y^{2} = 4 a x,

y^{2} = 4 a x

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