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 point of contact lies on the curve $xy=k{a}^2$, where $k=$

• A. $2$
• B. $3$
• C. $4$
• D. $5$

Answer 1

Answer: (C)

$4$

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 contact.

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 tangents to (2) at $(\alpha ,\beta )$ is

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

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

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

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