Sol. The equation of all conics whose axes are parallel to the coordinate axes and centres at the origin is Ax 2 + By 2 = 1... (i)

Let the centre be (a, 0). Then, equation (i) becomes

A(x –a) 2 + By 2 = 1 ... (ii)

If it touches y-axis at the origin, then

Aa 2 + By 2 = 1 [Putting x = 0 in (ii)]

must have equal roots i.e. Aa 2 + By 2 = 1 must have equal roots.

∴ Aa 2 = 1 ⇒ A =

Putting A = in (ii), we get

– + By 2 = 0 ... (iii)

as the equation of the family of conics satisfying the given properties.

Differentiating (iii) w.r.t. x, we get

– + 2By = 0

⇒ – + By = 0 ... (iv)

Differentiating (iv) w.r.t.x, we get

+ B + By = 0 ... (v)

Eliminating , and B from (iii), (iv) and (v), we get

= 0

⇒ + (2x – x 2 ) = 0

⇒ x 2 y + = 0, which is the required differential equation.