Question

The maximum area of the rectangle whose sides pass through the angular points of a given the rectangle is of sides a and b is

• A. (1/2) (ab)²
• B. (1/2) (a + b)
• C. (1/2) (a + b)²
• D. ) None of these

Answer 1

Answer: (C)

(1/2) (a + b)²

Answer 2

Let ABCD be the given rectangle of sides a and b and EFGH be any rectangle, whose sides pass through A, B, C, D

A = area EFGH = (b sin q + a cos q) (a sin q + b cos q)

= ab + (a² + b²) sin q cos q

dA/dq = (a² + b²) cos 2q so dA/dq = 0

Ž q = p/4

Ž = – 2 (a² + b²) sin 2q, so $\left. \frac { d ^ { 2 } A } { d \theta ^ { 2 } } \right| _ { \theta = \pi / 4 }$ < 0

Hence A_max = (1/2) (a + b)².