The director is excited about the progress you're already making in modeling the actor's blocking on stage. He's so excited, in fact, that he is already debating how to re-position Clara on the stage.
Suppose the director wants to move Clara one foot closer to center stage and two feet further downstage. He also wants to try a smaller spotlight on Clara--one that projects just 2 feet of light around the actor.
What is the equation of the circle that the spotlight will now fill around Clara? To find out, you can use the following steps.
| Identify the coordinates of the center of the circle (h,k). |
| Identify the length of the radius (r). The given measure could be some other value related to the radius, such as diameter, circumference, or area. |
| Substitute the coordinates (h,k) and the measure of the radius (r) into the general equation for a circle. |
First, let's use what we know about Clara's new position to find the center of the circle. If Clara's original coordinates were (3,4) and the director wants to move her one foot closer to center stage and two feet farther downstage, how can we determine the new center?
Subtract 1 from the x-coordinate, and subtract 2 from the y-coordinate. Clara's new position is (2,2).
The distance from the edge of the smaller spotlight's beam to Clara's position in the center is 2 feet. This distance is also the radius of the spotlight's circle. Now that we know the coordinates of the center (h,k) and the radius r, we can substitute these values into the general equation of a circle. Try writing this equation correctly in your notebook. Then, check your answer below.
\(\mathsf{(x-2)^2 + (y-2)^2 = 4}\)
See if you can figure out some additional equations of circles. Answer each question below in your notebook before clicking the button to check your answer.
Example 1
Example 2
Example 3
Example 4
Write the equation of a circle with center C (-1, 2) and a radius of 4 units.
| Start with the general equation of a circle. | \(\mathsf{(x-h)^2 + (y-k)^2 = r^2}\) |
| Identify h, k, and r. | h is -1, k is 2, and r is 4 |
| Substitute h, k, and r in the equation. | \(\mathsf{(x - (-1))^2 + (y - 2)^2 = 4^2}\) |
| Simplify. | \(\mathsf{(x + 1)^2 + (y - 2)^2 = 16}\) |
Write the equation of a circle with center C (4,0) and a radius of 5 units.
| Start with the general equation of a circle. | \(\mathsf{(x-h)^2 + (y-k)^2 = r^2}\) |
| Identify h, k, and r. | h is 4, k is 0, and r is 5 |
| Substitute h, k, and r in the equation. | \(\mathsf{(x - 4)^2 + (y - 0)^2 = 5^2}\) |
| Simplify. | \(\mathsf{(x - 4)^2 + y^2 = 25}\) |
Write the equation of a circle with center C (2, 5) and a radius of 4 units.
\(\mathsf{(x-2)^2 + (y-5)^2 = 16}\)
Write the equation of a circle with center C (-1, 2) and a radius of 5 units.
\(\mathsf{(x+1)^2 + (y-2)^2 = 25}\)
Question
How could you write the equation of a circle if you were given the circle's diameter instead of its radius?
Divide the diameter by two to find the radius because the diameter is twice the radius.
Question
How could you write the equation if you were given a point on the circle instead of the radius?
Substitute the coordinates of the center of the circle in the distance formula to solve for the measure of the radius.
