Question

The differential equation of all circles passing through the origin and having their centres on the X-axis is ______.

Options

• x² = `y^2 + xy dy/dx`

• x² = `y^2 + 3xy dy/dx`

• y² = `x^2 + 2xy dy/dx`

• y² = `x^2 - 2xy dy/dx`

Answer 1

The differential equation of all circles passing through the origin and having their centres on the X-axis is `underlinebb(y^2 = x^2 + 2xy dy/dx)`.

Explanation:

Let the equation of the circle be

x² + y² – 2gx = 0  ...(i)

Differentiating w.r.t. x,

`2x + 2y dy/dx - 2g` = 0

`\implies` 2g = `(2x + 2y dy/dx)`

Putting 2g in equation (i),

`x^2 + y^2 - (2x + 2y dy/dx)x` = 0

`\implies` y² = `x^2 + 2xy dy/dx`