Find the differential equation of all parabolas whose axes are parallel to the x-axis an having latus rectum a.

Answer

`a(d^(2)y)/(dx^(2))+2((dy)/(dx))^(2)=0`

Explanation

Equation of parabola whose axes is parallel to x-axis and having latus rectum 'a' is `(y-beta)^(2)=a(x-alpha)`
Here we have two effective constants `alpha` and `beta`
So it is required to differentiate twice.
Differentiating both sides,
We get `2(y-beta)(dy)/(dx)=a`..............(2)
Differentiating (2), w.r.t. x
`rArr 2(y-beta)(d^(2)y)/(dx^(2))+2((dy)/(dx))^(2)=0`............(3)
Eliminating `beta` from (2) and (3(,
`rArr a.(d^(2)y)/(dx^(2))+2((dy)/(dx))^(3)=0`,
Which is required differential equation.