Question

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

Options

•  a² + b²

•  a + b

•  a² − b²

• \[\sqrt{a2 + b2}\]

   

Answer 1

We have to find the distance betweenA` (a cos theta + b sin theta , 0 ) " and " B(0, a sin theta - b cos theta )` . 

In general, the distance between A(x₁ , y ₁)  and B(x₂ , y₂) is given by,

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

So,

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

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

But according to the trigonometric identity,

`sin^2  theta + cos^2  theta = 1` 

Therefore,

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