let P is variable point on line Y=4.tangents are drawn to circle x²+y² =4 from P touching circle at A and B.if parallelogram PAQB is completed the prove that locus of Q will be (y+4)(x² +y²)=2y²

Dear piyush

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

     
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