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P is a variable point on the line y=4. Tangents PA and PB are drawn to the circle x 2 +y 2 =4. A point Q is taken such that parallelogram PAQB is completed. Prove that locus of Q is (y+4)(x 2 +y 2 )=2y 2 .

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 .

Grade:11

1 Answers

Arun
25750 Points
5 years ago
Dear Ishaan
 
 

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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