In each of the question, form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

y^{2} = a(b^{2} – x^{2})

The given equation is y^{2} = a(b^{2} – x^{2})

Now, differentiating both sides w.r.t x, we get,

⇒ 2yy’ = -2ax

⇒ yy’ = -ax -------(1)

Now, again differentiating both sides, we get,

y’.y’ +yy’’ = -a

⇒ (y’)^{2} + yy” = -a --------(2)

Now, dividing equation (2) by (1), we get,

⇒ xyy” + x(y’)^{2} – yy” = 0

Therefore, the required differential equation is xyy” + x(y’)^{2} – yy” = 0.

2