Question

If the two parabolas $y^2=4a(x-2)$ and $x^2=4a(y-3)$ touch each other, then their point of contact lies on a

• A. circle
• B. line
• C. ellipse
• D. hyperbola

Answer 1

The correct option is D hyperbola

Let $P(x_1,y_1)$ be the point of contact of the two given parabolas

$y^2=4a(x-2)$ ...(1)

and $x^2=4a(y-3)$ ...(2)

Equation of tangent at $P$ to (1) is

$y{y}_1=2a(x+x_1)-8a\Rightarrow 2ax-y_{1}y+(2a{x}_1-8a)=0$ ...(3)

Equation of tangent at $P$ to (2) is

$x{x}_1=2a(y+y_1)-12a\Rightarrow x_{1}x-2ay-(2a{y}_1-12a)=0$

Since (3) and (4) represent the same line,

$\therefore \frac{2a}{x_1}=\frac{-y_1}{-2a}\Rightarrow x_1{y}_1=4a^2$

$\therefore$ point of contact $(x_1,y_1)$ lies on the curve $xy=4a^2$ which is hyperbola.