Q14 of 48 Page 99

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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