A tangent to the parabola $x^{2} =$ 4ay meets the hyperbola $x y = k^{2}$ in two points P & Q, then find the locus of the middle point of $P Q$ :

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Solution

Equation of tangent to parabola $x^{2} = 4 a y$ is $y - m x + a m^{2} = 0$ ....(i)
Let mid point of $P Q$ is $(x_{1}, y_{1})$
Then equation of $P Q$ is
$x y_{1} + y x_{1} = 2 k^{2}$ ....(ii)
On comparing (i) & (ii)
$\frac{x_{1}}{1} = \frac{y_{1}}{- m} = \frac{2 k^{2}}{a m^{2}}$
$\Rightarrow x_{1} = \frac{2 k^{2}}{a m^{2}}$ .....(iii)
$\Rightarrow y_{1} = \frac{2 k^{2}}{a m}$ .....(iv)
using (iii) & (iv) eliminate $m$ to get required locus,
$y^{2} = \frac{2 k^{2}}{a} x$ which is a parabola.

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A tangent to the parabola x2=4ay meets the hyperbola xy=k2 in two points P & Q, then find the locus of the middle point of PQ :

A tangent to the parabola $x^{2} =$ 4ay meets the hyperbola $x y = k^{2}$ in two points P & Q, then find the locus of the middle point of $P Q$ :