Question

Sol. [Key Concept: Tangents from the origin to $x^{2} + y^{2} + 2gx + 2fy + f^{2} = 0$ are perpendicular when the origin lies on the director circle, giving $g = \pm \ f$ (the $f^{2}$ makes it touch an axis).]

$$\begin{matrix} \sqrt{g^{2} + f^{2}} & \ = \frac{\sqrt{g^{2} + f^{2} - f^{2}}}{sin45^{\circ}} \\ \Rightarrow \ g^{2} + f^{2} & \ = 2(g)^{2} \\ g^{2} & \ = f^{2} \Rightarrow g = \pm f \end{matrix}$$