Find the locus of the mid points of all chords of the parabola $y^{2} = 4 a x$ which are drawn through the vertex.

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Solution

R.E.F image
consider the round AB as
shown, sine it pass

through fous . so $[t_{1} t_{2} = - 1]$

Mid pt . $(\frac{a}{2} (t_{1}^{2} + t_{2}^{1}), a (t_{1} + t_{2}))$

$x = \frac{a}{2} (t_{1}^{2} + t_{2}^{2})$ $[y = a (t_{1} + t_{1})]$ ---(i)

$\frac{2 x}{a} = t_{1}^{2} + t_{2}^{2}$

$\frac{2 x}{a} = (t_{1}^{2} + t_{2}^{2})$

$\frac{2 x}{a} = (t_{1} + t_{2})^{2} - 2 t_{1} t_{1}$

$\frac{2 x}{a} = \frac{y^{2}}{a^{2}} + 2$ -- $[(t_{1} + t_{2}) = \frac{y}{a}, t_{1} t_{2} = - 1]$

$2 x = (\frac{y^{2} + 2 a^{2}}{a})$

$[y^{2} = 2 a (x - a)]$

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