# Orthogonal trajectory for $y = ax^2$ will be ($a$ is parameter) Option 1) $x^2 + y^2 = C$ Option 2) $x^2 +2 y^2 = C$ Option 3) $x^2 - y^2 = C$ Option 4) $xy = c$

H Himanshu

As we have learnt,

Orthogonal Trajectory -

Any curve which cuts every member of a given family of curves at right angles

- wherein

First of all we need differential equation for given family of curves. So, on differentiating we get -

$\frac{\mathrm{d} y}{\mathrm{d} x} = 2ax\Rightarrow \frac{\mathrm{d} y}{\mathrm{d} x} = 2x\times \frac{y}{x^2} \Rightarrow \frac{\mathrm{d} y}{\mathrm{d} x} = \frac{2y}{x}$

Now if we replace $\frac{\mathrm{d} y}{\mathrm{d} x}$ with $-\frac{\mathrm{d} x}{\mathrm{d} y}$, the resulting equation will be differential equation, for required orthogonal trajectory.

So we get -

$-\frac{\mathrm{d} x}{\mathrm{d} y} = \frac{2y}{x} \Rightarrow xdx +2ydy = 0$

Now, Integrating we get -

$\frac{x^2}{2} + y^2 = c\Rightarrow x^2 + 2y^2 = 2c \Rightarrow x^2 + 2y^2 = c$

Option 1)

$x^2 + y^2 = C$

Option 2)

$x^2 +2 y^2 = C$

Option 3)

$x^2 - y^2 = C$

Option 4)

$xy = c$

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