Question

Parabolas $y^2=4a(x-c_1)$ and $x^2=4a(y-c_2)$, where $c_1$ and $c_2$ are variable, are such that they touch each other. Locus of their point of contact is

• A. $xy=2a^2$
• B. $xy=4a^2$
• C. $xy=a^2$
• D. none of these

Answer 1

Answer: (B)

$xy=4a^2$

Let $P(h,k)$ be the point of contact.
Tangent at $P$ will be the common tangent.
Equation of first parabola is
$y^2=4a(x-c_1)$
$\Rightarrow \frac{dy}{dx}=\frac{2a}{y}$
Slope of tangent at P $=\frac{2a}{k}$ .....(1)
Equation of second parabola is
$x^2=4a(y-c_2)$
$\Rightarrow \frac{dy}{dx}=\frac{x}{2a}$
Slope of tangent at P $=\frac{h}{2a}$ .....(2)
$\frac{2a}{k}=\frac{h}{2a}$
$\Rightarrow hk=4a^2$
or,$xy=4a^2$