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Circle that touches a given line at a given point

This type of question, asked 3 times: 2021 Paper 7, 27 Jun 2022, 27 Jan 2024. All 3 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2021 Paper 7

    74
    The line 2x−y+1=02x-y+ 1 = 0 is a tangent to the circle at the point (2,5)(2,5) and the centre of the circle lies on x−2y=4x- 2y= 4. Then, the radius of the circle is:

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  2. Question 2

    27 Jun 2022

    179
    Let a circle CC of radius 5 lie below the xx-axis. The line L1=4x+3y+2L_{1}= 4x + 3y + 2 passes through the centre PP of the circle CC and intersects the line L2:3x−4y−11=0L_{2}:3x - 4y - 11 = 0 at QQ. The line L2L_{2} touches CC at the point QQ. Then the distance of PP from the line 5x−12y+51=05x - 12y + 51 = 0 is

  3. Question 3

    27 Jan 2024

    178
    Consider a circle (x−α)2+(y−β)2=50(x-\alpha)^{2}+ (y-\beta)^{2}= 50, where α,β>0\alpha,\beta> 0. If the circle touches the line y+x=0y+x= 0 at the point PP, whose distance from the origin is 424\sqrt{2} , then (α+β)2(\alpha+\beta)^{2} is equal to........