Question

Sol. [Key Concept: The other diagonal of the square is the perpendicular bisector of the intercept, of equal length.]

Other vertices are

$$\left( \frac{a}{2} + \frac{\sqrt{a^{2} + b^{2}}}{2}\frac{b}{\sqrt{a^{2} + b^{2}}}, \right.\ \left. \ \frac{b}{2} + \frac{\sqrt{a^{2} + b^{2}}}{2}\frac{a}{\sqrt{a^{2} + b^{2}}} \right)$$

and $\left( \frac{a}{2} - \frac{\sqrt{a^{2} + b^{2}}}{2}\frac{b}{\sqrt{a^{2} + b^{2}}} \right.\ $, $\frac{b}{2} - \frac{\sqrt{a^{2} + b^{2}}}{2}\frac{a}{\sqrt{a^{2} + b^{2}}}$)

$$\begin{matrix} \\ \Rightarrow \ \left( \frac{a + b}{2},\frac{a + b}{2} \right)\text{~}\text{and}\text{~}\left( \frac{a - b}{2},\frac{b - a}{2} \right) \end{matrix}$$

✔ (a) $(\frac{a + b}{2},\frac{a + b}{2})$ is one extremity.

✘ (b) not equidistant from the centre.

✔ (c) $(\frac{a - b}{2},\frac{b - a}{2})$ is the other extremity.

✘ (d) not a vertex of the square.