- Graph a circle.
- Find the equation of a circle in the x−y plane.
- Find the radius and center, given the equation of a circle and vice versa.
- Find the equation of a circle, given the center and a point on the circle.
Graphing a Circle in the Coordinate Plane
Recall that the definition of a circle is the set of all points that are the same distance from the center. This definition can be used to find an equation of a circle in the coordinate plane.
Solution: The center is (0, 0). It’s radius is the square root of 9, or 3. Plot the center, and then go out 3 units in every direction and connect them to form a circle.
If you square both sides of this equation, then we would have the standard equation of a circle.
Example 2: Find the center and radius of the following circles.
When finding the center of a circle always take the opposite sign of what the value is in the equation.
Example 3: Find the equation of the circle below.
Solution: First locate the center. Draw in the horizontal and vertical diameters to see where they intersect.
Finding the Equation of a Circle
a) (8, -3)
b) (-2, -2)
(8, -3) is not on the circle
(-2, -2) is on the circle
Example 5: Find the equation of the circle with center (4, -1) and passes through (-1, 2).
Solution: First plug in the center to the standard equation.
- Questions 1-4 are similar to Examples 1 and 2.
- Questions 5-8 are similar to Example 3.
- Questions 9-11 are similar to Example 4.
- Questions 12-15 are similar to Example 5.
Find the center and radius of each circle. Then, graph each circle.
Find the equation of the circles below.
- Is (-7, 3) on (x+1)2+(y−6)2=45?
- Is (9, -1) on (x−2)2+(y−2)2=60?
- Is (-4, -3) on (x+3)2+(y−3)2=37?
- Is (5, -3) on (x+1)2+(y−6)2=45?
Find the equation of the circle with the given center and point on the circle.
- center: (2, 3), point: (-4, -1)
- center: (10, 0), point: (5, 2)
- center: (-3, 8), point: (7, -2)
- center: (6, -6), point: (-9, 4)