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

 

Answers (1)
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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