Question

If x/a cos θ + y/b sin θ = 1 and x/a sin θ – y/b cos θ = 1, prove that (x2/ a2 + y2/b2) = 2.

Answer 1

Answer:

x

cosθ+

b

y

sinθ=1−−−−−(1)

a

x

sinθ−

b

y

cosθ=1−−−−−(2)

Squaring both the equations and then adding ,

\bold{[\frac{x}{a}cos\theta+\frac{y}{b}sin\theta]^2+[\frac{x}{a}sin\theta-\frac{y}{b}cos\theta]^2=1^2+1^2}[

a

x

cosθ+

b

y

sinθ]

2

+[

a

x

sinθ−

b

y

cosθ]

2

=1

2

+1

2

x²/a² cos²θ + y²/b² sin²θ + 2xy/ab sinθ.cosθ + x²/a² sin²θ + y²/b² cos²θ - 2xy/ab sinθ.cosθ = 2

⇒x²/a² (cos²θ + sin²θ) + y²/b² (sin²θ + cos²θ ) = 2

⇒x²/a² × 1 + y²/b² × 1 = 2 [ ∵ sin²x + cos²x = 1 from trigonometric identities ]

∴ x²/a² + y²/b² = 2 , hence proved