Question

Find the distance between the points (a sin theta , a cos theta )and a cos theta ,-a sin theta )​

Answer 1

FORMULA TO BE USED

$distance \ formula = \sqrt{(x_{1}-x_{2} ) ^{2} + (y_{1} -y_{2}) ^{2} }$

GIVEN

points given are ( a sinθ , a cosθ )  and ( a cosθ , - a sinθ )

so,

x₁ = a sin θ

x₂ = a cos θ

y₁ = a cos θ

y₂ = - a sin θ

SOLUTION

$distance = \sqrt{(a sin\theta - a cos\theta)^2 + ( a cos\theta + a sin\theta)^2} \\\\distance = \sqrt{a^2sin^2\theta+ a^2cos^2\theta-2a^2sin\theta cos\theta+ a^2sin^2\theta + a^2cos^2\theta + 2 a^2sin\theta cos\theta} \\\\distance = \sqrt{2( a^2sin^2\theta + a^2cos^2\theta)} \; \\\\\\distance = \sqrt{2a^2 ( sin^2\theta+cos^2\theta)} \\\\( by \ identity \ sin^2\theta + cos^2\theta = 1 )\\\\distance = \sqrt{2a^2(1)} \\\\distance =( \sqrt{2} \ a ) \ units$

ANSWER

distance between the given points will be √2 a units.