Question

The equation to the orthogonal trajectories of the system of parabolas $y=a{x}^2$ is

• A. $\frac{x^2}{2}+y^2=c$
• B. $x^2+\frac{y^2}{2}=c$
• C. $\frac{x^2}{2}-y^2=c$
• D. $x^2-\frac{y^2}{2}=c$

Answer 1

The correct option is A $\frac{x^2}{2}+y^2=c$

Given the equation of the parabola is $y=a{x}^2$......(1).

Now, differentiating both sides of (1) we get, $\frac{dy}{dx}=2ax$ or, $a=\frac{y^{\prime }}{2x}$.

Using this in equation (1) we get,

$y=\frac{x{y}^{\prime }}{2}$......(2).

This is the differential equation corresponding to the parabola (1).

Now, the differential equation of the orthogonal trajectory of (1) is

$y=-\frac{x}{2y^{\prime }}$ [ Replacing $y^{\prime }$ in (2) by $-\frac{1}{y^{\prime }}$]

or, $2ydy+xdx=0$

Now integrating both sides we get,

$y^2+\frac{x^2}{2}=c$ [ Where $c$ is integrating constant]

This is the required equation of orthogonal trajectory to (1).