4.0 Orthogonal trajectory

Any curve which cuts every member of a given family of curve at right angle is called an orthogonal trajectory of a given system of curve. Steps to find the orthogonal trajectory are:

(i) Let $f(x,y,c)=0$ be the equation of the given family of curves, where $c$ is an arbitrary constant.

(ii) Differentiate the given equation with respect to $x$ and eliminate $c$ to form a differential equation.

(iii) Replace $\frac{{dy}}{{dx}}$ by $ - \frac{{dx}}{{dy}}$ in the equation obtained after (ii).

(iv) On solving the differential equation obtained in (iii), we get the required orthogonal trajectory.

Question 16. Find the orthogonal trajectory of ${y^2} = 4ax$ ($a$ being the parameter).

Solution: The given equation is $${y^2} = 4ax\quad ...(1)$$

Differentiate the given equation with respect to $x$, we get $$2y\frac{{dy}}{{dx}} = 4a\quad ...(2)$$

Eliminate the parameter $a$ using $(1)$ and $(2)$, we get $${y^2} = 2y\frac{{dy}}{{dx}}x$$

Replace $\frac{{dy}}{{dx}}$ by $ - \frac{{dx}}{{dy}}$, we get $$y = 2\left( { - \frac{{dx}}{{dy}}} \right)x \Rightarrow 2xdx + ydy = 0 $$ Integrating each term, we get $$\Rightarrow 2\int {xdx} + \int {ydy} = 0 $$$$ {x^2} + \frac{{{y^2}}}{2} = c $$$$ 2{x^2} + {y^2} = 2c $$ which is the required orthogonal trajectories.