Question

Two parabolas $y^2=4a(x-{\lambda }_1)$ and $x^2=4a(y-{\lambda }_2)$ always touch each other $({\lambda }_1,{\lambda }_2$ being variable parameters). Then their point of contact lies on a

• A. straight line
• B. circle
• C. parabola
• D. hyperbola

Answer 1

The correct option is D hyperbola

The parabola $y^2=4a(x-{\lambda }_1)$ and $x^2=4a(y-{\lambda }_2)$ always touch each other (${\lambda }_1,{\lambda }_2$ being variable parameter ). then their point of contact lies on :

$y{y}_1=4a(\frac{x}{2}+\frac{x_1}{2}-{\lambda }_1)2ax-y_{1}y+2a{x}_1-4a{\lambda }_1=\mathrm{0.......}\left(1\right)x{x}_1=4a(\frac{y}{2}+\frac{y_1}{2}-{\lambda }_2)x{x}_1-2ay-2a{y}_1+4a{\lambda }_2=\mathrm{0.......}\left(2\right)$

(1) and (2) represents the same line

$\frac{2a}{x_1}=-\frac{y_1}{2a}=\frac{2a{x}_1-4a{\lambda }_1}{-2a{y}_1+4a{\lambda }_2}=k-4a^2=x_1{y}_1$

Replace

$x_1\to x{y}_1\to y-4a^2\to c^{2}xy=c^2$

Which is a hyperbola.