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Focii of Ellipse

Equation of a...

Question

Equation of a line is given by $y + 2 a t = t (x - a t^{2})$ , $t$ being the parameter. Find the locus of the point of intersection of the lines which are at right angles.

y^{2} = - a (x - 3 a)

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x^{2} = a (y - 3 a)

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y^{2} = a (x - 3 a)

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x^{2} = - a (y - 3 a)

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Solution

The correct option is B $y^{2} = a (x - 3 a)$
$y + 2 a t = t a - a t^{3}$
Slope is $= t$
Let is passes through $P (h, k)$
$∴ k + 2 a t = t h - a t^{3}$
$\Rightarrow a t^{3} + t (2 a - h) + k = 0$ ......(1)
$t_{1} t_{2} t_{3} = - \frac{k}{a} {t_{1} t_{2} = - 1}$
$t_{3} = \frac{k}{a}$
Substituting $t_{3}$ in (1) we can get the locus
$y^{2} = a (x - 3 a)$

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