Let's continue to set up the opening scene of Porgy and Bess by adding to our sketch. You've seen how the actors can be represented as points on the coordinate plane. How do you think the spotlights will be shown?
The shape of each spotlight should form a circle with the actor as the center of the circle.
The director wants to know what happens when each spotlight is illuminated to its maximum width of 6 feet. What is the boundary of the circle that will enclose Clara? Will it shine on other actors nearby?
To find the boundary of the spotlight circle, we need to define the distance between Clara (that is, the center of her circle) and any point on the circle surrounding her. When you want to find the distance between points in the coordinate plane, you should be thinking, of course, about the distance formula.
\(\mathsf{\sqrt{(x_2 -x_1 )^2+(y_2 -y_1 )^2}}\)=d
Because Clara is at the center of the circle, the distance between Clara at point (3,4) and any point (x, y) on the circle is a radius. If the circle has a width of 6 feet, its radius is 3 feet, so d in this case is 3.
In your notebook, try applying the distance formula to this scenario. Insert the information you know into the distance formula and see if you can derive the equation of the circle. When you're done, click the button below to check your answer.
| Replace (x1, y1) with (3,4) and (x2, y2) with (x, y) and d with 3. | \(\mathsf{\sqrt{(x-3)^2+(y-4)^2}}\)=3 |
| Square each side of the equation. | \(\mathsf{{(x-3)^2+(y-4)^2}}\)=\(\mathsf{3^2}\) |
| Simplify. | \(\mathsf{{(x-3)^2+(y-4)^2}}\)=9 |
The equation above defines a circle with center at (3,4) and a radius of 3 units. You can generalize from this example to identify a standard equation that works for any circle. In the coordinate plane, a circle is represented as follows:
The equation of a circle with center (h, k) and a radius of r units is \(\mathsf{{(x-h)^2 + (y-k)^2}}\) = \(\mathsf{r^2}\)
Using this general equation of a circle, you can find the equation of a particular circle if you know the center of the circle and the length of its radius.
