P AND Q ARE TWO POINTS ? AND 2? ON THE ELLIPSE x²/a²+y²/b²=1 .SHOW THAT THE LOCUS OF THE MIDPOINT OF PQ HAS THE EQUATION

 (x²/a²+y²/b²)( 4x²/a²+4y²/b²-3)=x/3

P = (a costheta , bsintheta )

Q = ( acos2theta , b sin2theta )

mid point of PQ = ( h , k )

h = a( cos theta + cos2 theta ) / 2

k = b( sintheta + sin 2 theta ) /2

eliminate theta from these 2 equations

I think u made a typo error.The RHS must be x/a.

Continuing with what asish has done,2x/a=cosk+cos2k and 2y/b=sink+sin2k,

Squaring and adding,we get,4x²/a²+4y²/b²=2+2cosk and x²/a²+y²/b²=(1+cosk)/2

Subtract 3 from eqn1 and multiply the above eqns,the RHS is 1/2(cosk+1)(2cosk-1)=1/2(2cos²k+cosk-1)=1/2(cosk+cos2k)=x/a