Question

If x sin θ = y cos θ = `(2z  tan θ)/(1 - tan^2 θ)`, then 4z²(x² + y²) is equal to ______.

Options

• (x² + y²)³

• (x² – y²)³

• (x² – y²)²

• (x² + y²)²

Answer 1

If x sin θ = y cos θ = `(2z  tan θ)/(1 - tan^2 θ)`, then 4z²(x² + y²) is equal to (x² – y²)².

Explanation:

Given, `x/("cosec"θ) = y/secθ = z/(cot 2θ)` = k (say)

∴ 4z² (x² + y²) = 4k² cot² 2θ [k²cosec²θ + k² sec² θ]

= `4k^4 cot^2 2θ (1/(sin^2θ cos^2θ))`  ...[∵ sin² θ + cos² θ = 1]

= `(4k^4)/4 ((cos^2θ - sin^2θ)/(sin^2θ cos^2θ))^2`

= (k²cosec² θ – k² sec² θ)²

= (x² – y²)²