The video on the previous page showed you how to transform a slightly different equation of a circle into a form you recognized and could use to graph the circle. (Remember, if your circle is written in the standard equation form, it is much easier to identify the key parts of the circle.) See if you can apply these steps and the other skills you have learned in this lesson. Work each of these problems on your own, and then click the Show Me button to check your work.
Problem 1
Problem 2
Problem 3
Find the coordinates of the center and the radius of a circle with the equation:
\(\mathsf{x^2 + y^2 - 6x +20y +108=0.}\)
1. Regroup the terms:\(\small\mathsf{x^2 - 6x +y^2 + 20y = -108.}\)
2. Complete the square: \(\small\mathsf{x^2 - 6x + 9 + y^2 +20y +100 = -108 + 9 + 100}\)
3. Simplify, rewriting perfect squares as factors: \(\small\mathsf{(x-3)^2 + (y+10)^2 = 1}\)
4. Identify the parts from the manipulated equation: h = 3, k = -10, r=1
The center is (3, -10), and the measure of the radius is 1 (1 is the square root of 1).
Meteorologists have been tracking hurricane Ariel, which is expected to come ashore. Residents who live within a certain radius of Ariel's predicted path are required to evacuate. If an equation of the circle that presents the evacuated area is given by (x + 55)2 + (y - 14)2 = 1024, find the coordinates of the center and the measure of the radius of the evacuated area.
The center is (-55, 14) and the radius is 32.
A smartphone app called Find Me Food will find give you a list of restaurants within 7 miles of your current location. Assuming your current location is represented by the coordinates (0,0), write the equation of the circle that marks the boundary of the area the app will show you.
Since your location is the center of the circle and the distance the restaurants are from you is bounded at a radius of 7 miles, insert your coordinates as (h,k) and r = 7 in the general equation of a circle:
\(\mathsf{(x-0)^2 + (y-0)^2 = 7^2 \text{ or } x^2 + y^2 = 49.}\)
