Question

The elimination of θ from xcos θ - ysinθ = 2 and xsinθ + ycosθ = 4 will give:
1. x² - y² = 20
2. 3x² + y² = 20
3. x² + y² = 20
4. 3x² - y² = 20

Answer 1

Correct Answer - Option 3 : x² + y² = 20

Given : 

xcosθ - ysinθ = 2 

xsinθ + ycosθ = 4 

Formula used: 

sin²θ + cos²θ = 1

(a + b)² = a² + b² + 2ab

Calculation: 

xcosθ - ysinθ = 2 

On squaring we get,

⇒ x²cos²θ + y²sin²θ – 2(xcosθ)(ysinθ) = 4      ….(i)

xsinθ + ycosθ = 4 

On squaring we get,

⇒ x²sin²θ + y²cos²θ + 2(ycosθ)(xsinθ) = 16      ….(ii)

On adding eqns (i) and (ii)

⇒ x²cos²θ + y²sin²θ – 2(xcosθ)(ysinθ) + x²sin²θ + y²cos²θ + 2(ycosθ)(xsinθ) = 4 + 16

⇒ x²(cos²θ + sin²θ) + y²(sin²θ + cos²θ) = 20

⇒ x² + y² = 20

∴ The correct answer is x² + y² = 20