The locus of the midpoints of the focal chords of the parabola y 2-4 ax is A- y2-2 a2 x-aB- y2-a2 x-aC- y2-2 ax-aD- y2-2 ax-a

The correct option is D 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 yy_1 2ax=y^2_1 ...

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The locus of ...

Question

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

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Solution

The correct option is B
Given equation of the parabola is $y^{2} = 4 a x$ .
We know that the ends of focal chords are $(a t^{2}, 2 a t)$ and $(\frac{a}{t^{2}}, \frac{- 2 a}{t})$
Let $(h, k)$ be the mid point of the chord, then
$(h, k) = (\frac{a t^{2} + \frac{a}{t^{2}}}{2}, \frac{2 a t - \frac{2 a}{t}}{2})$
$\Rightarrow k = \frac{2 a t - \frac{2 a}{t}}{2}$
$∴ k = a (t - \frac{1}{t})$ .......(i)
Similar way, $h = \frac{a t^{2} + \frac{a}{t^{2}}}{2}$
$\Rightarrow 2 h = a (t^{2} + \frac{1}{t^{2}})$
$\Rightarrow 2 h = a [(t - \frac{1}{t})^{2} + 2]$
$\Rightarrow 2 h = a [(\frac{k}{a})^{2} + 2]$
$\Rightarrow 2 h = \frac{k^{2} + 2 a^{2}}{a}$
$\Rightarrow 2 h a = k^{2} + 2 a^{2}$
$\Rightarrow k^{2} = 2 h a - 2 a^{2}$
$∴ k^{2} = 2 a (h - a)$
Hence, the locus of the midpoints of the focal chords of the given parabola is $y^{2} = 2 a (x - a)$

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The locus of the midpoints of the focal chords of the parabola y2=4ax is

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