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Math Help - chord

  1. #1
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    chord

    the origin O is the midpoint of a chord PQ of the circle (x - 6)^2 + (y + 3)^2 = 65, show the gradient of the chord PQ is 2

    could someone please show me how you would figure this out, thanks
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  2. #2
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    Equation of chord of a circle when its midpoint is given is S_1=S_11 where S is the equation the circle.

    Given equation is
    <br />
x^2 + y^2 - 12x +6y -20 = 0
    Equation of a chord when its midpoint (x_1,y_1)
    is  xx_1 + yy_1 - 6(x+x_1) + 3(y+y_1) - 20 = x_1^2 + y_1^2 - 12x_1 + 6y_1 -20
    Here midpoint is 0rigin
    Therefore
     x(0) + y(0) - 6(x+0) + 3(y +0) - 20 = 0^2 + 0^2 -12(0) + 6(0) - 20

    Therefore equation of chord is
     6x - 3y = 0 whose slope is -6/-3 =2
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  3. #3
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    i don't think i understand that, is that called differentiation? because i haven't started doing that yet. is there a more simple way to do it?
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  4. #4
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    Okay I will suggest another method.

    Let
    y= mx be chord to circle
    centre is C (6,-3)
    Mid pt is T(0,0)

    Slope of CT = 0-(-3)/0-6 = -1/2

    Since CT is perpendicular to the chord, slope of the chord = m= 2

    the equation becomes y = 2x

    therefore gradient = 2
    Last edited by nolanfan; October 18th 2009 at 06:45 AM.
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  5. #5
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    i think there's something wrong here, the book asked find the length of the chord PQ and the answer was 4\sqrt{5} but the length of the centre to the origin is 3\sqrt{5} which i think is what you were talking about. the centre (6, -3) is labelled as C by the way, so i'm not sure if it has anything to do with PQ
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  6. #6
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    Well, I made a mistake by taking the midpoint (0,0) to be P randomly. I've changed the midpoint now to T. I didn't notice that the ends of the chord were P and Q. So let T (0,0) be the midpoint of PQ. Now the length of CT is 3\sqrt5. But  \Delta CTP is a right angled triangle. therefore,
    By pythogarus theorem
    <br />
CP^2 = CT^2 + TP^2
    where TP = 1/2 length of chord.

    Also CP = radius = \sqrt65

    \sqrt65^2 = 3\sqrt5^2  + TP^2

    TP^2 = 20

    TP = 2\sqrt5

    Length of chord = 2 x TP = 4\sqrt5
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