Equation of a Circle Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

medium
Find the equation of the circle that has a diameter with endpoints (−1,4)(-1, 4) and (5,−2)(5, -2).

Solution

  1. 1
    The center is the midpoint: (−1+52,4+(−2)2)=(2,1)\left(\frac{-1+5}{2}, \frac{4+(-2)}{2}\right) = (2, 1). The diameter length is (5−(−1))2+(−2−4)2=36+36=62\sqrt{(5-(-1))^2 + (-2-4)^2} = \sqrt{36+36} = 6\sqrt{2}, so r=32r = 3\sqrt{2}.
  2. 2
    (x−2)2+(y−1)2=(32)2=18(x-2)^2 + (y-1)^2 = (3\sqrt{2})^2 = 18.

Answer

(x−2)2+(y−1)2=18(x-2)^2 + (y-1)^2 = 18
When given the endpoints of a diameter, the center is the midpoint and the radius is half the diameter length. The midpoint formula and distance formula are the key tools here.

About Equation of a Circle

The standard form equation (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2 describes a circle with center (h,k)(h, k) and radius rr in the coordinate plane.

Learn more about Equation of a Circle →

More Equation of a Circle Examples

Example 1 easy

Write the equation of the circle with center [formula] and radius [formula].

Example 2 medium

Find the center and radius of the circle [formula].

Example 4 hard

Determine whether the circles [formula] and [formula] intersect, and find the number of intersection

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