Question

If the two parabolas $y^2=4a(x-k_1)$ and $x^2=4a(y-k_2)$ always touch each other, $k_1$ and $k_2$ being variable parameters, then their point of contact lies on the curve :

• A. $xy=a^2$
• B. $xy=2a^2$
• C. $xy=4a^2$
• D. $xy=3a^2$

Answer 1

Answer: (C)

$xy=4a^2$

Given:

$H_1:y^2=4a(x-k_1)$

$H_2:x^2=4a(y-k_2)$

Let point of contact of $H_1$ and $H_2$ be $(h,k)$

Tangent of both curves have same slope

$\Rightarrow \frac{4a}{2k}=\frac{2h}{4a}$

$\Rightarrow 4hk=16a^2$

$\Rightarrow hk=4a^2$

$\therefore xy=4a^2$ is the required locus.

Hence, option C.