P is a variable point on the line y=4.Tangents PA and PB are drawn to the circle x²+y²=4. A point Q is taken such that parallelogram PAQB is completed.

Prove that locus of Q is

(y+4)(x²+y²)=2y² .

chord of contact of circle from P is

 ax +4y =4 .............1

equation of chord bisected at a point O is

 x(a+h)/2  + y (k+4)/2 -4 = {(a+h)/2}²  + {(k+4)/2}²  -4

 x(a+h)/2  + y (k+4)/2  = {(a+h)/2}²  + {(k+4)/2}²   ....................2

equation 1 and 2 represent same line AB so

 (a+h)/2a   = (k+4)/2*4    =  [{(a+h)/2}²  + {(k+4)/2}²  ]/4

 so  (a+h)/2a   = (k+4)/2*4

    1+ h/a  = k/4 +1

   a = 4h/k

 and   (k+4)/2*4    =  [{(a+h)/2}²  + {(k+4)/2}²  ]/4

         ( k+4)/2    =    {(4h/k+h)/2}²  + {(k+4)/2}²

        ( k+4)/2    =    {(4h/k+h)/2}²  + {(k+4)/2}²

          ( k+4)/2    =   h²/k² {(k+4)/2}²  + {(k+4)/2}² 

            (k+4)(h² +k²)=2k²