Equation of a Circle Math Example 2

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

Example 2

medium
Find the center and radius of the circle x2+y2−6x+4y−12=0x^2 + y^2 - 6x + 4y - 12 = 0.

Solution

  1. 1
    Group xx and yy terms: (x2−6x)+(y2+4y)=12(x^2 - 6x) + (y^2 + 4y) = 12.
  2. 2
    Complete the square for xx: x2−6x+9=(x−3)2x^2 - 6x + 9 = (x-3)^2. Add 99 to both sides.
  3. 3
    Complete the square for yy: y2+4y+4=(y+2)2y^2 + 4y + 4 = (y+2)^2. Add 44 to both sides.
  4. 4
    (x−3)2+(y+2)2=12+9+4=25(x-3)^2 + (y+2)^2 = 12 + 9 + 4 = 25.
  5. 5
    Center: (3,−2)(3, -2), radius: 25=5\sqrt{25} = 5.

Answer

Center (3,−2),r=5\text{Center } (3, -2), \quad r = 5
Completing the square converts the general form x2+y2+Dx+Ey+F=0x^2 + y^2 + Dx + Ey + F = 0 into standard form (x−h)2+(y−k)2=r2(x-h)^2 + (y-k)^2 = r^2. This reveals the center and radius directly.

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 3 medium

Find the equation of the circle that has a diameter with endpoints [formula] and [formula].

Example 4 hard

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

← All Equation of a Circle Examples Equation of a Circle Concept