Form the differential equation representing the family of curves given by `(x-a)^2+2y^2=a^2`, where a is an arbitrary constant.

Explanation

Given, family of curve, `(x-a)^(2)+2y^(2)=a^(2)`, where `a` is constant.
`implies x^(2)-2ax+2y^(2)=0`……`(1)`
Differentiate w.r.t.x,
`2x-2a+4y(dy)/(dx)=0`…….`(2)`
Multiply equation `(2)` by `x` and subtract equation `(1)` from it,
`x(2x-2a+4y(dy)/(dx))-(x^(2)-2ax+2y^(2))=0`
`2x^(2)-2ax+4xy(dy)/(dx)-x^(2)+2ax-2y^(2)=0`
`implies 4xy(dy)/(dx)+x^(2)-2y^(2)=0`
`implies (dy)/(dx)=(2y^(2)-x^(2))/(4xy)`
which is the required differential equation.