Question

Find the orthogonal trajectory of `y^2=4a x` (a being the parameter).

Answer 1

`y^(2)=4ax`…………..(1)
`therefore 2y(dy)/(dx)=4a`……………………..(2)
Eliminating a from equation (1) and (2), we get
`y^(2)=2y(dy)/(dx)x`
Replacing `(dy)/(dx)dy-(dx)/(dy)`, we get
`y=2(-(dx)/(dy))x`
`2xdx+ydy=0`
Integrating both sides, we get
`x^(2)+y^(2)/(2)=c`
`2x^(2)+y^(2)=2c`
Which is the required orthogonal trajectory.