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. None of these

Answer 1

The correct option is C $xy=4a^2$

Given parabolas are $y^2=4a(x-k_1)$ ...(1)

and $x^2=4a(y-k_2)$ ...(2)

Let $(\alpha ,\beta )$ be their point of conatct.

Equation of tangent to (1) at $(\alpha ,\beta )$ is

$\beta y=2a(x-k_1+\alpha )\Rightarrow 2ax-\beta y=2a(k_1-\alpha )$ ...(3)

Equation of tangent to (2) at $(\alpha ,\beta )$ is

$\alpha x=2a(y-k_2+\beta )\Rightarrow \alpha x-2ay=2a(\beta -k_2)$ ...(4)

Since (3) and (4) are identical. comparing coefficients of $x$ and $y$ is (3) and (4), we get

$\frac{2a}{\alpha }=\frac{\beta }{2a}\Rightarrow \alpha \beta =4a^2$

i.e. the point contact $(\alpha ,\beta )$ lies on the curve $xy=4a^2$