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maths-urgently needed

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find the equation of circle
if the line 2x - y +1 = 0 touches the circle at the point (2,5) and the center of the circle lies on the line x+y -9 =0

  • maths-urgently needed -

    If the line touches the circle, then the line from (2,5) to the center is perpendicular to the line y=2x+1, so it has slope -1/2.

    So, the line through (2,5) with slope -1/2 is

    y-5 = -1/2 (x-2)
    y = -1/2 x + 6

    This line passes through the center of the circle, so

    -1/2 x + 6 = 9-x
    x=6

    (6,3) is the center of the circle.
    The radius is the distance from (6,3) to (2,5) = √20

    The circle is thus

    (x-6)^2 + (y-3)^2 = 20

    Just as a final check:
    The distance from (6,3) to the line 2x-y+1 = 0 is |2*6 - 3 + 1|/√5 = 10/√5 = 2√5 = √20

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