Find the locus of the middle points of the chord of contact of tangents to the circle x2+y2=1 drawn from the points on the circle x2+y2+4x+6y+9=0 .

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Please find below the solution to the asked query:

x2+y2=1Equation of chord of contact from x1,y1 isT=0xx1+yy1-1=0....iwhere x1,y1 lies on x2+y2+4x+6y+9=0i.e. x12+y12+4x1+6y1+9=0....iiLet mid point of chord of contact be h,kEquation of chord having mid point h,k isT=S1xh+yk-1=h2+k2-1xh+yk-h2+k2=0....iiii and iii represent same line, hence compare coefficientsx1h+y1k=1h2+k2x1,y1=hh2+k2,kh2+k2Put in ii to get locus.hh2+k22+kh2+k22+4hh2+k2+6kh2+k2+9=0h2+k2+4hh2+k2+6kh2+k2+9h2+k22=0x2+y2+4xx2+y2+6yx2+y2+9x2+y22=0

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