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Circle touching one axis at a given point and cutting a given intercept on the other

This type of question, asked 3 times: 2021 Paper 17, 28 Jan 2025, 6 Apr 2026 S2. All 3 questions below, oldest first.

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  1. Question 1

    2021 Paper 17

    77
    Consider a circle CC which touches the yy-axis at (0,6)(0,6) and cuts off an intercept 656\sqrt{5} on the x-axis. Then the radius of the circle CC is equal to:

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  2. Question 2

    28 Jan 2025

    58
    Let the equation of the circle, which touches xx-axis at the point (a,0),a>0(a,0),a > 0 and cuts off an intercept of length bb on yy-axis be x2+y2−αx+βy+γ=0x^{2}+y^{2}- \alpha x + \beta y + \gamma = 0. If the circle lies below x-axis, then the ordered pair (2a,b2)\left( 2a,b^{2} \right) is equal to :

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  3. Question 3

    6 Apr 2026 S2

    60
    Let C be a circle having centre in the first quadrant and touching the x -axis at a distance 3 units from the origin. If the circle C has an intercept of length 636\sqrt{3} on yy-axis, then the length of the chord of the circle CC on the line x−y=3x-y= 3 is :

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