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Math Help - Circles and Chords

  1. #1
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    Circles and Chords

    Having a bit of trouble with this one. Can anyone help me out?

    Many thanks.

    Q.
    C is a circle with centre (-1, -4). The midpoint of a chord of length 2\sqrt{5} is (2, 0). Find the length of the radius of C.

    Attempt: Perpendicular distance [bc] from centre c to x-axis is 4,
    Let |ab| = 1/2 length of chord on x-axis = \sqrt{5},
    |ac|^2=(\sqrt{5})^2+4^2=21\rightarrow |ac|=\sqrt{21}= radius,

    Ans:
    (From text book): Radius = \sqrt{30}
    Last edited by GrigOrig99; May 19th 2013 at 11:03 AM.
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  2. #2
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    Re: Circles and Chords

    Quote Originally Posted by GrigOrig99 View Post
    Q.[/B] C is a circle with centre (-1, -4). The midpoint of a chord of length 2\sqrt{5} is (2, 0). Find the length of the radius of C.

    Attempt: Perpendicular distance [bc] from centre c to x-axis is 4
    So I assume b is (-1, 0).

    Quote Originally Posted by GrigOrig99 View Post
    Let |ab| = 1/2 length of chord on x-axis = \sqrt{5}
    I am not sure what the "length of chord on x-axis" is: the projection of the chord to the x-axis? Why are you interested in it since it is not known that the chord is parallel to the x-axis? Also, "Let |ab| = \sqrt{5}" is ambiguous. Does this mean that a = (-1-\sqrt{5},0) or (-1+\sqrt{5},0), or maybe a does not lie on the x-axis at all?

    Hint: The line from the circle center to the chord center is perpendicular to the chord. Using this fact, find a right triangle with a radius as the hypotenuse.
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  3. #3
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    Re: Circles and Chords

    Apologies for the late reply.

    The problem with the 'length of chord on x-axis' is the part that threw me, in that all previous questions placed the chords on either the x or y-axis, and I was assuming the same was true, here.

    Let |ab| = 1/2 length of chord = \frac{2\sqrt{5}}{2}}=\sqrt{5},
    With centre c = (-1, 4), calculate perpendicular distance to cord via (2, 0) = \sqrt{(-1-2)^2+(-4-0)^2}=\sqrt{25}=5,
    |ac|^2=\sqrt{5}^2+5^2=30\rightarrow |ac|=\sqrt{30}
    Equation of a circle: (x+1)^2+(y+4)^2=\sqrt{30}^2\rightarrow x^2+y^2+2x+8y-13=0

    Thanks again.
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