You draw a circle that is centered at . You measure the diameter of the circle to be 18 units. Does the point lie on the circle?
Guidance
When a circle is centered at the origin (as in the last concept), the equation is . If we rewrite this equation, using the center, it would look like . Extending this idea to any point as the center, we would have , where is the center.
Example A
Find the center and radius of and graph.
Solution: Using the general equation above, the center would be and the radius is or 4. To graph, plot the center and then go out 4 units up, down, to the left, and to the right.
Example B
Find the equation of the circle with center and radius 5.
Solution: Plug in the center and radius to the equation and simplify.
Example C
Find the equation of the circle with center and is on the circle.
Solution: In this example, we are not given the radius. To find the radius, we must use the distance formula, .
Therefore, the equation of this circle is or .
Intro Problem Revisit In this lesson, you learned the equation of a circle that is centered somewhere other than the origin is , where is the center.
We are given that the center is , so and . We are also given the diameter of the circle, but we need the radius. Recall that the radius is half the diameter, so .
If we plug these values into the equation for the circle, we get:
Now to find if the point lies on the circle we substitute 4 for x and 5 for y and see if the equation holds true.
Therefore, the point does not lie on the circle.
Guided Practice
1. Graph and find the center and radius.
2. Find the equation of the circle with center and is on the circle.
3. The endpoints of a diameter of a circle are and . Find the equation.
Answers
1. The center is and the radius is 5.
2. Use the distance formula to find the radius.
The equation of this circle is .
3. In this problem, we are not given the center or radius. We can find the length of the diameter using the distance formula and then divide it by 2.
Now, we need to find the center. Use the midpoint formula with the endpoints.
Therefore, the equation is .
Vocabulary
- Standard Form (of a Circle)
- , where is the center and is the radius.
Explore More
For questions 1-4, match the equation with the graph.
Graph the following circles. Find the center and radius.
Find the equation of the circle, given the information below.
- center: radius: 7
- center: radius:
- center: point on circle:
- center: point on circle:
- diameter endpoints: and
- diameter endpoints: and
- Is on the circle ? How do you know?
- Challenge Using the steps from #16 in the previous concept, find the equation of the tangent line to the circle with center and the point of tangency is .
- Extension Rewrite the equation of the circle, in standard form by completing the square for both the and terms. Then, find the center and radius.