Question

Parabola $y^2=4a(x-c_1)$ and $x^2=4a(y-c_2)$ where $c_1$ and $c_2$ are variables, touch each other. Locus of their point of contact is?

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

Answer 1

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

$y^2=4a(x-c_1)$

and $x^2=4a(y-c_2)$ where $C_1$ and $C_2$ are variables, are such that they touch each other.

Locus of their point of contact is:

Equation of parabolas are : $y^2=4a(x-C_1)$

and $x^2=4a(y-C_2)$

where $C_1$ and $C_2$ are variables.

Here both the given curves touch each other at $2$ points. At the point of contact $(x,y)$, then the slope of both curves at $(x,y)$ are the same.

$y^2=4a(x-a)$

differentiate wrt $x$

$2y{y}^{\prime }=4a......\left(1\right)$

$X^2=4A(y-(2\left)\right)$ differentiate wrt $x$

$2x=4a{y}^{\prime }.....\left(2\right)$

solving $\left(1\right)$ and $\left(2\right)$,

$xy=4a^2$

hence locus of the point of contact is $xy=4a^2$