# If the coordinates of P and Q are (a cos θ, b sin θ) and (–a sin θ, b cos θ) respectively, then prove that OP2 + OQ2 = a2 + b2, where O is the origin.

We have P (a cos θ, b sin θ)

Q (–a sin θ, b cos θ)

We know that distance of a point A (x,y) from origin O (0, 0) is given as OA =

Using the above formula,

OP =

=

OP2 =

OQ =

=

OQ2 =

Now, OP2 + OQ2 = a2 cos2θ + b2 sin2θ + a2 sin2θ + b2 cos2θ

= a2 (cos2θ +sin2θ) + b2 (sin2θ + cos2θ)

We know that, cos2θ +sin2θ =1

OP2 + OQ2 = a2 + b2

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