Question

Given the interval 0<θ<π/2. Find the angle θ which is formed by the line y = -2x+4 and y = 3x-3?

Answer 1

Answer:

Concept:

sin 2A = 2 sin A × cos A

cos 2A = cos2 A – sin2 A

Calculation:

Given: p = X cos θ - Y sin θ and q = x sin θ + Y cos θ.

⇒ p2 = x2 cos2 θ + Y2 sin2 θ – 2 XY cos θ sin θ ...1)

⇒ q2 = x2 cos2 θ + Y2 sin2 θ + 2 XY cos θ sin θ ...2)

⇒ 4pq = 4 [X2 cos θ sin θ + XY cos2 θ – XY sin2 θ – Y2 sin θ cos θ] ...3)

By adding (1), (2) and (3), we get

⇒ p2 + 4pq + q2 = X2 [1 + 2 sin 2θ] + Y2 [1 - 2 sin 2θ] + 4 XY cos 2θ

As it is given that, p2 + 4pq + q2 = AX2 + BY2

⇒ X2 [1 + 2 sin 2θ] + Y2 [1 - 2 sin 2θ] + 4 XY cos 2θ = AX2 + BY2

⇒ A = 1 + 2 sin 2θ, B = 1 - 2 sin 2θ and cos 2θ = 0.

⇒ θ = π/4, A = 3 and B = - 1.