Question

P is a point on the segment joining the feet of two vertical poles of heights a and b. The angles of elevation of the tops of the poles from P are 45° each. Then, the square of the distance between the tops of the poles is

(a) \(\frac{a^2\,+\,b^2}{2}\)

(b) a² + b²

(c) 2(a² + b²)

(d) 4(a² + b²)

Answer 1

(c) 2(a² + b²)

Let AD and BC be the two pillar of heights a and b respectively.

∠DPA = 45°, ∠CPB = 45°

Required distance = CD.

Draw DE || AB meeting BC in E

∴ AB = DE

In Δ APD,

tan 45° = \(\frac{a}{AP}\)

⇒ AP  = a

In Δ BPC,

tan 45° = \(\frac{b}{PB}\)

⇒ PB = b

∴ AB = AP + PB = a + b

⇒ DE = a + b

CE = BC – BE = BC – AD = b – a

In Δ DEC,

DC² = DE² + CE² = (a + b)² + (b – a)²

= 2 (a² + b²).