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Thread: Equation of a Circle

  1. #1
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    Equation of a Circle

    Find the equation of a circle whose centre falls on the line y = 6 - 2x and which passes through the points A(-2,0) and B(4,0)
    Since the y-value of both points A and B are 0, I can gather from this question that the diameter is 6 units and so the radius is 3 units. But I can't solve any further.
    Any help would be greatly appreciated, thank you!
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  2. #2
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    Well the equation of a circle with radius size $\displaystyle r$ is $\displaystyle \displaystyle (x-h)^2+(y-k)^2=r^2$ where $\displaystyle (h,k)$ is its centre.

    Now use the information given (substitute in some points) to solve for $\displaystyle r, h, k$
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  3. #3
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    Hello, caramelcake!

    Find the equation of the circle whose center is on $\displaystyle y \:=\:6-2x$
    and which passes through $\displaystyle A(\text{-}2,0)$ and $\displaystyle B(4,0).$

    Since the $\displaystyle y$-value of both points $\displaystyle A$ and $\displaystyle B$ are 0,
    I can gather that the diameter is 6 units. . This is not true!

    Did you make a sketch?


    Code:
               \    |
                \   * * *
                *\  |       *
              *   \ | :       *
             *     \| :        *
                    * :
            *       |\:C        *
            *       | o         *
            *       | :\        *
                    | : \
             *      | :  \     *
          - - o - - + - - \ - o - - - - -
             -2 *   | :    \* 4
                    * * *   \
                    | :

    The $\displaystyle x$-intercepts of the circle are: $\displaystyle A(\text{-}2,0)$ and $\displaystyle B(4,0)$

    The center lies on the vertical line midway between the intercepts: .$\displaystyle x \:=\:1$

    The center also lies on the line $\displaystyle y \:=\:6-2x$

    . . The center is the intersection of the two lines: .$\displaystyle C(1,4)$

    The radius is the distance $\displaystyle \overline{CA}\!:\;5 $


    Therefore: .$\displaystyle (x-1)^2 + (y-4)^2 \:=\:5^2$
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  4. #4
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    Thank you pickslides and Soroban for your input Terribly sorry for the late reply, I didn't log in in time to check the second reply before submitting my homework >.<
    Either way, I have already completed it so here's my answer:
    x-coordinate of centre is $\displaystyle \frac{-2+4}{2}$$\displaystyle =1$
    $\displaystyle y = 6 - 2 = 4$
    radius of circle = $\displaystyle \sqrt{(1 + 2)^2 + (4 - 0)^2}
    $
    $\displaystyle = \sqrt{25}$
    $\displaystyle = 5 units$
    equation of the circle is
    $\displaystyle (x + 1)^2 + (y - 4)^2 = 5^2$
    $\displaystyle x^2 - 2x +1 + y^2 - 8y + 16 = 25$
    $\displaystyle x^2 + y^2 - 2x - 8y -8 = 0$
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