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Circle touching a parabola at a given point

This type of question, asked 2 times: 14 Jun 2022, 25 Jul 2022. All 2 questions below, oldest first.

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  1. Question 1

    14 Jun 2022

    65
    Let x2+y2+Ax+By+C=0x^{2}+y^{2}+Ax+By+C= 0 be a circle passing through (0,6)(0,6) and touching the parabola y=x2y=x^{2} at (2,4)(2,4). Then A+CA + C is equal to

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  2. Question 2

    25 Jul 2022

    89
    The sum of diameters of the circles that touch (i) the parabola 75x2=64(5y−3)75x^{2}= 64(5y - 3) at the point (85,65)\left( \frac{8}{5},\frac{6}{5} \right) and (ii) the yy-axis, is equal to