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Q. A circle is drawn such that it passes through the point 0, 1 and touches the parabola y = 2x2 at (1, 2) tangents are drawn from 
the points p (8/3, 10/3) to touch the circle at a and b. The diameter of the circle circumscribing the triangle pab is ? 

Dear Student,
Please find below the solution to the asked query:

Let equation of cirlce be x2+y2+2gx+2fy+c=0Circle passes through 0,1, hence0+1+0+2f+c=02f+c=-1..iAs circle touches parabola at 1,2, hence 1,2 lies on circle1+4+2g+4f+c=02g+4f+c=-5...iiAs circle and parabola will have common tangent at point of contact.Slope of tangent ofparabola at 1,2=Slope of tangent of circle at 1,2y=2x2Differentiatedydx=4xdydx1,2=4×1=4x2+y2+2gx+2fy+c=0Differentiate2x+2y.dydx+2g+2f.dydx=02x+2y.dydx1,2+2g+2f.dydx1,2=02×1+2×2×4+2g+2f×4=02g+8f=-18g+4f=-9...iiiSolving i,ii,iii, we can get value of g and f.Centre of cirlce is C-g,-fAs we already know that P83,103, A,B,C are concyclic, and PAC=PBC=90°, hence Pwill be diameter which you caneasily find using C-g,-f and P83,103

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