LESSON 10–8 Equations of Circles

Five-Minute Check (over Lesson 10–7) TEKS Then/Now New Vocabulary Key Concept: Equation of a Circle in Standard Form Example 1: Write an Equation Using the Center and Radius Example 2: Write an Equation Using the Center and a Point Example 3: Graph a Circle Example 4: Real-World Example: Use Three Points to Write an Equation Example 5: Intersections with Circles Lesson Menu

Find x. A. 1 B. 2 C. 3 D. 4 5-Minute Check 1

Find x. A. 1 B. 2 C. 3 D. 4 5-Minute Check 2

Find x. A. 2 B. 4 C. 6 D. 8 5-Minute Check 3

Find x. A. 10 B. 9 C. 8 D. 7 5-Minute Check 4

Find x in the figure. A. 14 B. C. D. 5-Minute Check 5

Mathematical Processes G.1(E), G.1(G) Targeted TEKS G.2(B) Derive and use the distance, slope, and midpoint formulas to verify geometric relationships, including congruence of segments and parallelism or perpendicularity of pairs of lines. G.12(E) Show that the equation of a circle with center at the origin and radius r is x 2 + y 2 = r 2 and determine the equation for the graph of a circle with radius r and center (h, k), (x - h)2 + (y - k)2 = r 2. Mathematical Processes G.1(E), G.1(G) TEKS

You wrote equations of lines using information about their graphs. Write the equation of a circle. Graph a circle on the coordinate plane. Then/Now

compound locus Vocabulary

Concept

(x – h)2 + (y – k)2 = r 2 Equation of circle Write an Equation Using the Center and Radius A. Write the equation of the circle with a center at (3, –3) and a radius of 6. (x – h)2 + (y – k)2 = r 2 Equation of circle (x – 3)2 + (y – (–3))2 = 62 Substitution (x – 3)2 + (y + 3)2 = 36 Simplify. Answer: (x – 3)2 + (y + 3)2 = 36 Example 1

B. Write the equation of the circle graphed to the right. Write an Equation Using the Center and Radius B. Write the equation of the circle graphed to the right. The center is at (1, 3) and the radius is 2. (x – h)2 + (y – k)2 = r 2 Equation of circle (x – 1)2 + (y – 3)2 = 22 Substitution (x – 1)2 + (y – 3)2 = 4 Simplify. Answer: (x – 1)2 + (y – 3)2 = 4 Example 1

A. Write the equation of the circle with a center at (2, –4) and a radius of 4. A. (x – 2)2 + (y + 4)2 = 4 B. (x + 2)2 + (y – 4)2 = 4 C. (x – 2)2 + (y + 4)2 = 16 D. (x + 2)2 + (y – 4)2 = 16 Example 1

B. Write the equation of the circle graphed to the right. A. x2 + (y + 3)2 = 3 B. x2 + (y – 3)2 = 3 C. x2 + (y + 3)2 = 9 D. x2 + (y – 3)2 = 9 Example 1

Step 1 Find the distance between the points to determine the radius. Write an Equation Using the Center and a Point Write the equation of the circle that has its center at (–3, –2) and passes through (1, –2). Step 1 Find the distance between the points to determine the radius. Distance Formula (x1, y1) = (–3, –2) and (x2, y2) = (1, –2) Simplify. Example 2

Step 2 Write the equation using h = –3, k = –2, and r = 4. Write an Equation Using the Center and a Point Step 2 Write the equation using h = –3, k = –2, and r = 4. (x – h)2 + (y – k)2 = r 2 Equation of circle (x – (–3))2 + (y – (–2))2 = 42 Substitution (x + 3)2 + (y + 2)2 = 16 Simplify. Answer: (x + 3)2 + (y + 2)2 = 16 Example 2

Write the equation of the circle that has its center at (–1, 0) and passes through (3, 0). A. (x + 1)2 + y2 = 16 B. (x – 1)2 + y2 = 16 C. (x + 1)2 + y2 = 4 D. (x – 1)2 + y2 = 16 Example 2

Write the equation in standard form by completing the square. Graph a Circle The equation of a circle is x2 – 4x + y2 + 6y = –9. State the coordinates of the center and the measure of the radius. Then graph the equation. Write the equation in standard form by completing the square. x2 – 4x + y2 + 6y = –9 Original equation x2 – 4x + 4 + y2 + 6y + 9 = –9 + 4 + 9 Complete the squares. (x – 2)2 + (y + 3)2 = 4 Factor and simplify. (x – 2)2 + [y – (–3)]2 = 22 Write +3 as – (–3) and 4 as 22. Example 3

With the equation now in standard form, you can identify h, k, and r. Graph a Circle With the equation now in standard form, you can identify h, k, and r. (x – 2)2 + [y – (–3)]2 = 22 (x – h)2 + [y – k]2 = r2 Answer: So, h = 2, y = –3, and r = 2. The center is at (2, –3), and the radius is 2. Example 3

Which of the following is the graph of x2 + y2 –10y = 0? A. B. C. D. Example 3

Analyze You are given three points that lie on a circle. Use Three Points to Write an Equation ELECTRICITY Strategically located substations are extremely important in the transmission and distribution of a power company’s electric supply. Suppose three substations are modeled by the points D(3, 6), E(–1, 1), and F(3, –4). Determine the location of a town equidistant from all three substations, and write an equation for the circle. Analyze You are given three points that lie on a circle. Formulate Graph ΔDEF. Construct the perpendicular bisectors of two sides to locate the center, which is the location of the tower. Find the length of a radius. Use the center and radius to write an equation. Example 4

Use Three Points to Write an Equation Determine Graph ΔDEF and construct the perpendicular bisectors of two sides. Example 4

Use Three Points to Write an Equation The center, C, appears to be at (4, 1). This is the location of the tower. Find r by using the Distance Formula with the center and any of the three points. Write an equation. Example 4

the circle and the given information. Use Three Points to Write an Equation Answer: The location of a town equidistant from all three substations is at (4,1). The equation for the circle is (x – 4)2 + (y – 1)2 = 26. Justify You can verify the location of the center by finding the equations of the two bisectors and solving a system of equations. You can verify the radius by finding the distance between the center and another of the three points on the circle. Evaluate Analyze the graph to find the relationship between the radius and center of the circle and the given information. Example 4

AMUSEMENT PARKS The designer of an amusement park wants to place a food court equidistant from the roller coaster located at (4, 1), the Ferris wheel located at (0, 1), and the boat ride located at (4, –3). Determine the location for the food court. A. (3, 0) B. (0, 0) C. (2, –1) D. (1, 0) Example 4

Find the point(s) of intersection between x2 + y2 = 32 and y = x + 8. Intersections with Circles Find the point(s) of intersection between x2 + y2 = 32 and y = x + 8. Graph these equations on the same coordinate plane. Example 5

x2 + y2 = 32 Equation of circle. Intersections with Circles There appears to be only one point of intersection. You can estimate this point on the graph to be at about (–4, 4). Use substitution to find the coordinates of this point algebraically. x2 + y2 = 32 Equation of circle. x2 + (x + 8)2 = 32 Substitute x + 8 for y. x2 + x2 + 16x + 64 = 32 Evaluate the square. 2x2 + 16x + 32 = 0 Simplify. x2 + 8x + 16 = 0 Divide each side by 2. (x + 4)2 = 0 Factor. x = –4 Take the square root of each side. Example 5

Use y = x + 8 to find the corresponding y-value. Intersections with Circles Use y = x + 8 to find the corresponding y-value. (–4) + 8 = 4 The point of intersection is (–4, 4). Answer: (–4, 4) Example 5

Find the points of intersection between x2 + y2 = 16 and y = –x. Example 5

LESSON 10–8 Equations of Circles