Question

Find the distance between (a cos theta,0) and (0,a sin theta).​

Answer 1

Answer :-

Distance between ( a cos θ, 0 ) and ( 0, sin θ ) is 'a' units.

Explanation :-

Let A( a cos θ, 0 ) and B( 0, a sin θ ) be the given points.

$\boxed{ \boxed{ \textsf{Distance between two points ( d )} = \sf \sqrt{( {x_2 - x_1) }^{2} + { {(y_2 - y_1) }}^{2} } }}$

$\rm \implies AB = \sqrt{ {(0 - acos \theta)^{2} + {(asin \theta - 0)}^{2} }}$

$\rm \implies AB = \sqrt{ {( - acos \theta)^{2} + {(asin \theta) }^{2} }}$

$\rm \implies AB = \sqrt{ {( - a)^{2} cos^{2} \theta + {a}^{2} sin^{2} \theta }}$

$\rm \implies AB = \sqrt{ {a^{2} cos^{2} \theta + {a}^{2} sin^{2} \theta }}$

$\rm \implies AB = \sqrt{ {a^{2}( cos^{2} \theta + sin^{2} \theta )}}$

$\rm \implies AB = \sqrt{ {a^{2}(1)}}$

$\boxed{ \boxed{ \sf \because cos^{2} \theta + sin^{2} \theta = 1}}$

$\rm \implies AB = \sqrt{ {a^{2}}}$

$\rm \implies AB = a \ \ units$

Hence, the distance between ( a cos θ, 0 ) and ( 0, sin θ ) is 'a' units.

Answer 2

Solution

Let the given points be P(a cos∅,0) and Q(0,a sin∅)

To find

Length of PQ

Using Distance Formula,

_______________________

PQ = √( - a cos∅)² + (asin∅)²

___________________

→ PQ = √ a²cos²∅ + a²sin²∅

__________________

→ PQ = √a²(sin²∅ + cos²∅)

But sin²∅ + cos²∅ = 1

→ PQ = √a²

→ PQ = a units

Thus,the distance between P and Q is "a" units