First slide

Parabola

Question

$If the chord of contact of tangents from a point P to the parabola y^{2} = 4 a x touches the parabola x^{2} = 4 b y, then the$ $locus of P is$

Moderate

A

$x y = a b$
B

$x y = 2 a b$
C

$x y = - 2 a b$
D

$x y = - a b$

Solution

$The chord of contact of the parabola y^{2} = 4 a x w.r.t. point P (x_{1}, y_{1}) is$
$y y_{1} = 2 a (x + x_{1}) - - - - - - - (1)$
$This line touches the parabola x^{2} = 4 b y .$
$Solving (1) with parabola, we have$
$x^{2} = 4 b [\frac{2 a}{y_{1}} (x + x_{1})]$
$or y_{1} x^{2} - 8 a b x - 8 a b x_{1} = 0$
According to the question, this equation must have equal roots.
Therefore, D = 0
$\begin{array}{ll} or 64 a^{2} b^{2} + 32 a b x_{1} y_{1} = 0 \\ or x_{1} y_{1} = - 2 a b or x y = - 2 a b \end{array}$
which is the required locus.

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