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Math Help - circle analytic geometry

  1. #1
    TMU
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    circle analytic geometry

    Hello there. I'm currently studying AS level maths in the UK and have been given a sheet of circle questions (This is not my strong point either).
    I'm hoping you can help with this .

    1) A circle passes through the points (2,0) and (8,0) and has the y axis as a tangent. Find the two possible equations.

    2) A(6,3) and B(10,1) are two points on a circle with centre (11,8).
    Calculate
    Part 1 - the co-ordinates of the mid-point of the chord
    Part 2 - the disbance of the chord AB from the centre of the circle
    Part 3 - the radius of the circle.

    Thanks a lot for your time and help, please
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  2. #2
    MHF Contributor
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    For question one:
    Let the center of the circle be (h, k).
    Since the circle is tangent to the y-axis, the point (0, k) is on the circle.
    And since both (2,0) and (8,0) are on the circle, they form an isosceles triangle with (h, k), which means that h = 5. Hence, (h, k) = (5, k). So the radius of the circle is 5.

    Solving for k by using the distance formula:
    \sqrt{(5-2)^2 + (k-0)^2} = 5

    \sqrt{3^2 + k^2} = 5

    9 + k^2 = 25

    k^2 = 16

    k = \pm4.

    Hence the two equations are:

    (x - 5)^2 + (y - 4)^2 = 25

    (x - 5)^2 + (y + 4)^2 = 25
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  3. #3
    TMU
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    Thanks so much for your help, actually understand that question well. Now the other question :S
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  4. #4
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    the chord is just the line connecting the two given points

    for two points (x_1,y_1) and (x_2,y_2)

    the midpoint of them would be (\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})

    so the point for part one is (\frac{6+10}{2},\frac{3+1}{2})=(8,2)

    parts two and three are just applying the distance formula, the first one is finding the distance from the center point (given) to the point I found in part 1 and the second is using it to find the distance between the center and any one of the two points they've given you

    the distance formula is
    d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}

    can you do it now?
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