Question

Sol. [Key Concept: A circle through the origin making intercepts $2a,2b$ on the axes has the form $x^{2} + y^{2} - 2ax - 2by = 0$ (signs vary); the tangent at $O$ is $ax + by = 0$.]

Eqn. of circle is

$$\left( x \pm a^{2} \right) + (y \pm b)^{2} = a^{2} + b^{2} $$or $(x \pm b)^{2} + (y \pm a)^{2} = a^{2} + b^{2}$

$$x^{2} + y^{2} \pm 2ax \pm 2by = 0 $$or $x^{2} + y^{2} \pm 2bx \pm 2ay = 0$

Eqn. of tangent origin is

$$ax \pm by = 0\ \text{~}\text{or}\text{~}\ bx \pm ay = 0$$