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Condition For Perpendicularity

Two parabolas...

Question

Two parabolas $y^{2} = 4 a (x - l_{1})$ and $x^{2} = 4 a (y - l_{2})$ always touch one another, the quantities $l_{1}$ and $l_{2}$ are both variable. Locus of their point of contact has the equation

A

x y = a^{2}

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B

x y = 2 a^{2}

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C

x y = 4 a^{2}

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D

None

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Solution

The correct option is C $x y = 4 a^{2}$
Let $P (x_{1}, y_{1})$ be point of contact of two parabola. tangents at P of the two parabolas are
$y y_{1} = 2 a (x + x_{1}) - 4 a l_{1}$ and
$x x_{1} = 2 a (x + y_{1}) - 4 a l_{2}$
$\Rightarrow 2 a x - y y_{1} = 2 a (2 l_{1} - x_{1})$ ......(i)
and $x x_{1} - 2 a y = 2 a (y_{1} - 2 l_{2})$ .....(ii)
Clearly (i) and (ii) represent same line
$∴ \frac{2 a}{x_{1}} = \frac{y_{1}}{2 a} \Rightarrow x_{1} y_{1} = 4 a^{2}$
Hence locus of P is $x y = 4 a^{2}$

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