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Given the endpoints of the diameter are A(x1, y1) = (2,4) and B(x2, y2) = (-2,6).

Mathematics

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Answer

x2+y2−10y+20=0x^2 + y^2 - 10y + 20 = 0

Step-by-step solution

4 steps

  1. Identify the coordinates of the endpoints of the diameter.

    Given the endpoints of the diameter are A(x1,y1)=(2,4)A(x_1, y_1) = (2,4) and B(x2,y2)=(−2,6)B(x_2, y_2) = (-2,6).

  2. Use the diameter form of the equation of a circle.

    The equation of a circle with endpoints of a diameter (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is given by:

    (x−x1)(x−x2)+(y−y1)(y−y2)=0(x-x_1)(x-x_2) + (y-y_1)(y-y_2) = 0

  3. Substitute the given coordinates into the equation.

    Substitute x1=2x_1=2, y1=4y_1=4, x2=−2x_2=-2, and y2=6y_2=6:

    (x−2)(x−(−2))+(y−4)(y−6)=0(x-2)(x-(-2)) + (y-4)(y-6) = 0 (x−2)(x+2)+(y−4)(y−6)=0(x-2)(x+2) + (y-4)(y-6) = 0

  4. Expand and simplify the equation.

    Expand the first product (x−2)(x+2)(x-2)(x+2):

    x2−(2)2=x2−4x^2 - (2)^2 = x^2 - 4

    Expand the second product (y−4)(y−6)(y-4)(y-6):

    y2−6y−4y+(−4)(−6)=y2−10y+24y^2 - 6y - 4y + (-4)(-6) = y^2 - 10y + 24

    Substitute these expanded forms back into the equation:

    (x2−4)+(y2−10y+24)=0(x^2 - 4) + (y^2 - 10y + 24) = 0 x2+y2−10y−4+24=0x^2 + y^2 - 10y - 4 + 24 = 0 x2+y2−10y+20=0x^2 + y^2 - 10y + 20 = 0
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