Question

The distance between the points (a cos θ + b sin θ, 0) and (0, a sin θ – b cos θ), is ______.

विकल्प

• a² + b²

• a² – b²

• `sqrt(a^2 + b^2)`

• `sqrt(a^2 - b^2)`

Answer 1

The distance between the points (a cos θ + b sin θ, 0) and (0, a sin θ – b cos θ), is `underlinebb(sqrt(a^2 + b^2))`.

Explanation:

Here, x₁ = a cos θ + b sin θ, y₁ = 0 and x₂ = 0, y₂ = a sin θ – b cos θ

∴ Distance = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`

= `sqrt((0 - a cos θ - b sin θ)^2 + (a sin θ - b cos θ - 0)^2`

= `sqrt((-1)^2 (a cos θ + b sin θ)^2 + (a sin θ - b cos θ)^2`

= `sqrt(a^2 cos^2θ + b^2 sin^2θ + 2ab cosθ sinθ + a^2 sin^2θ + b^2 cos^2θ - 2ab sinθ cosθ)`

= `sqrt(a^2(sin^2θ + cos^2θ) + b^2(sin^2θ + cos^2θ))`

= `sqrt(a^2 xx 1 + b^2 xx 1)`

= `sqrt(a^2 + b^2)`