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Circle from given tangent and normal lines

This type of question, asked 3 times: 28 Jun 2022, 30 Jan 2023, 13 Apr 2023. All 3 questions below, oldest first.

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  1. Question 1

    28 Jun 2022

    90
    Let the lines y+2x=11+77y+ 2x=\sqrt{11}+ 7\sqrt{7} and 2y+x=211+672y+x= 2\sqrt{11}+ 6\sqrt{7} be normal to a circle C:(x−h)2+(y−k)2=r2C:(x-h)^{2}+ (y-k)^{2}=r^{2}. If the line 11y−3x=5773+11\sqrt{11}y- 3x=\frac{5\sqrt{77}}{3}+ 11 is tangent to the circle CC, then the value of (5h−8k)2+5r2(5h- 8k)^{2}+ 5r^{2} is equal to

  2. Question 2

    30 Jan 2023

    67
    Let y=x+2,4y=3x+6y = x + 2,4y = 3x + 6 and 3y=4x+13y = 4x + 1 be three tangent lines to the circle (x−h)2+(y−(x - h)^{2}+ (y - k)2=r2k)^{2}=r^{2}. Then h+kh + k is equal to:

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  3. Question 3

    13 Apr 2023

    160
    Let the centre of a circle CC be (α,β)(\alpha,\beta) and its radius r<8r < 8. Let 3x+4y=243x + 4y = 24 and 3x−4y=323x - 4y = 32 be two tangents and 4x4x +3y=1+ 3y = 1 be a normal to CC. Then (α−β+r)(\alpha - \beta + r) is equal to

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