10-8 Equations of Circle
Write the equation of each circle.
1. center at (9, 0), radius 5
SOLUTION: 3. center at origin, passes through (2, 2)
SOLUTION: Find the distance between the points to determine the radius.
Write the equation using h = 0, k = 0, and r =
.
5. SOLUTION: Find the distance between the points to determine the radius.
Write the equation using h = 2, k = 1, and r = 2.
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10-8 Equations of Circle
5. SOLUTION: Find the distance between the points to determine the radius.
Write the equation using h = 2, k = 1, and r = 2.
For each circle with the given equation, state the coordinates of the center and the measure of the
radius. Then graph the equation.
2
2
7. x – 6x + y + 4y = 3
SOLUTION: The standard form of the equation of a circle with center at (h, k) and radius r is
Use completing the square to rewrite the given equation in standard form.
.
So, h = 3, k = –2, and r = 4. The center is at (3, –2), and the radius is 4.
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9. RADIOS Three radio towers are modeled by the points R(4, 5), S(8, 1), and T(–4, 1). Determine the location of
another tower equidistant from all three towers, and write an equation for the circle.
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10-8 Equations of Circle
For each circle with the given equation, state the coordinates of the center and the measure of the
radius. Then graph the equation.
2
2
7. x – 6x + y + 4y = 3
SOLUTION: The standard form of the equation of a circle with center at (h, k) and radius r is
Use completing the square to rewrite the given equation in standard form.
.
So, h = 3, k = –2, and r = 4. The center is at (3, –2), and the radius is 4.
9. RADIOS Three radio towers are modeled by the points R(4, 5), S(8, 1), and T(–4, 1). Determine the location of
another tower equidistant from all three towers, and write an equation for the circle.
SOLUTION: Step 1: You are given three points that lie on a circle. Graph triangle RST and construct the perpendicular bisectors
of two sides to locate the center of the circle. Find the radius and then use the center and radius to write an equation.
Construct the perpendicular bisectors of two sides. The center appears to be at (2, –1).
Step 2: Find the distance between the center and one of the points on the circle to determine the radius.
Step 3: Write the equation of the circle using h = 2, k = –1, and r =
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.
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10-8 Equations of Circle
9. RADIOS Three radio towers are modeled by the points R(4, 5), S(8, 1), and T(–4, 1). Determine the location of
another tower equidistant from all three towers, and write an equation for the circle.
SOLUTION: Step 1: You are given three points that lie on a circle. Graph triangle RST and construct the perpendicular bisectors
of two sides to locate the center of the circle. Find the radius and then use the center and radius to write an equation.
Construct the perpendicular bisectors of two sides. The center appears to be at (2, –1).
Step 2: Find the distance between the center and one of the points on the circle to determine the radius.
Step 3: Write the equation of the circle using h = 2, k = –1, and r =
.
The location of the tower equidistance from the other three is at (2, –1) and the equation for the circle is
.
Find the point(s) of intersection, if any, between each circle and line with the equations given.
11. SOLUTION: Graph these equations on the same coordinate plane. (x – 1)2 + y 2 = 4 is a circle with center (1, 0) and a radius
of 2. Draw a line through (0, 1) with a slope of 1 for y = x + 1. eSolutions Manual - Powered by Cognero
The points of intersection are solutions of both equations. You can estimate these points on the graph to be at
about (–1, 0) and (1, 2). Use substitution to find the coordinates of these points algebraically. Page 4
The location of the tower equidistance from the other three is at (2, –1) and the equation for the circle is
10-8 Equations of Circle
.
Find the point(s) of intersection, if any, between each circle and line with the equations given.
11. SOLUTION: Graph these equations on the same coordinate plane. (x – 1)2 + y 2 = 4 is a circle with center (1, 0) and a radius
of 2. Draw a line through (0, 1) with a slope of 1 for y = x + 1. The points of intersection are solutions of both equations. You can estimate these points on the graph to be at
about (–1, 0) and (1, 2). Use substitution to find the coordinates of these points algebraically. So, x = 1 or –1. Use the equation y = x + 1 to find the corresponding y-values.
When x = 1, y = 1 + 1 or 2.
When x = –1, y = –1 + 1 or 0.
Therefore, the points of intersection are (1, 2) and (–1, 0).
CCSS STRUCTURE Write the equation of each circle.
13. center at origin, radius 4
SOLUTION: 15. center at (–2, 0), diameter 16
SOLUTION: Since
the radius
is half
the diameter,
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r = (16) or 8.
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10-8 Equations of Circle
15. center at (–2, 0), diameter 16
SOLUTION: Since the radius is half the diameter, r = (16) or 8.
17. center at (–3, 6), passes through (0, 6)
SOLUTION: Find the distance between the points to determine the radius.
Write the equation of the circle using h = – 3, k = 6, and r = 3.
19. SOLUTION: The center is (–5, –1) and the radius is 3.
21. WEATHER A Doppler radar screen shows concentric rings around a storm. If the center of the radar screen is the
origin and each ring is 15 miles farther from the center, what is the equation of the third ring?
SOLUTION: The radius of the third ring would be 15 + 15 + 15 or 45.
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10-8 Equations of Circle
21. WEATHER A Doppler radar screen shows concentric rings around a storm. If the center of the radar screen is the
origin and each ring is 15 miles farther from the center, what is the equation of the third ring?
SOLUTION: The radius of the third ring would be 15 + 15 + 15 or 45.
Therefore, the equation of the third ring is x 2 + y 2 = 2025.
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