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`        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 .`
one year 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
```
one year ago
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