Question

P is a point on or inside the boundary of a square ABCD, then the minimum value of ŠPAB + ŠPBC + ŠPCD + ŠPDA is equal to-

• A. $\frac{3\pi}{4}$
• B. $\frac{\pi}{4}$
• C. $\frac{5\pi}{4}$
• D. $\frac{\pi}{2}$

Answer 1

Answer: (B)

$\frac{\pi}{4}$

Answer 2

Sol. Let the vertices of the square ABCD are 1, –1, i and –i in the Argand diagram.

Let P be represented by z then ŠPAB + ŠPBC + ŠPCD + ŠPDA

= arg $\left( \frac{z^{4} - 1}{4} \right)$= arg (z⁴ –1)

Since z⁴ also lies inside the square Ž $\frac{5\pi}{4}$ ³ arg (z⁴ – 1) ³ $\frac{3\pi}{4}$