Question

If `P(theta) and Q((pi)/(2)+theta)` are two points on the ellipse `(x^(2))/(9)+(y^(2))/(4)=1` then find the locus of midpoint of PQ

Answer 1

Correct Answer - `(4x^(2))/(9)+y^(2)=2`
We have points on the ellipse `P(3 cos theta,2 sin theta) and Q(-3 sin theta, 2 cos theta)`.
Let M(h,k) be the midpoint of PQ
`:.h=(3cos theta-3sin theta)/(2) andk=(2sin theta+2costheta)/(2)`
`:. costheta-sin theta=(2h)/(3) and k sin theta+cos theta`
Squaring and adding , we get
`(4h^(2))/(9)+k^(2)=2`
or `(4x^(2))/(9)+y^(2)=2`, which is the required locus.