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

Options

A. `x^2/a^2 + y^2/b^2 =1/2`
B. `x^2/a^2+y^2/b^2=4`
C. `x^2/a^2 + y^2/b^2=2`
D. None of these

Answer

A

Explanation

`because P-=(a cos theta , b sin theta )` and `Q-=(a cos (pi/2+theta), b sin (pi/2+theta))`
Let mid point of PQ is (x,y) , then `2x=a (cos theta - sin theta)`…(i) and `2y=b (sin theta + cos theta)`….(ii)
From eqation (i) , `(2x)/a = cos theta -sin theta` ...(iii)
And from equation (ii), `(2y)/b=sin theta = cos theta ` ...(iv)
Squaring and adding equation (iii) & equation (iv) , we get : `(4x^2)/a^2 + (4y^2)/b^2 =2 rArr x^2/a^2+y^2/b^2=1/2`