EQUATIONSOF

CIRCLES

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

Write an Equation Using the Center and Radius

B. Write the equation of the circle graphed to the right.

(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

The center is at (1, 3) and the radius is 2.

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

A. Write the equation of the circle with a center at (2, –4) and a radius of 4.

A. x2 + (y + 3)2 = 3

B. x2 + (y – 3)2 = 3

C. x2 + (y + 3)2 = 9

D. x2 + (y – 3)2 = 9

B. Write the equation of the circle graphed to the right.

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.

Write an Equation Using the Center and a Point

Step 2 Write the equation using h = –3, k = –2, andr = 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

A. (x + 1)2 + y2 = 16

B. (x – 1)2 + y2 = 16

C. (x + 1)2 + y2 = 4

D. (x – 1)2 + y2 = 16

Write the equation of the circle that has its center at (–1, 0) and passes through (3, 0).

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 equationx2 – 4x + 4 + y2 + 6y + 9 = –9 + 4 + 9 Complete the

squares.(x – 2)2 + (y + 3)2

= 4Factor and simplify.(x – 2)2 + [y – (–3)]2

= 22 Write +3 as – (–3) and 4 as 22.

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.

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

C. D.

Use Three Points to Write an Equation

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.Understand You are given three points that lie on a

circle.Plan Graph ΔDEF. Construct the perpendicular

bisectors of two sides to locate the center,which is the location of the tower. Find thelength of a radius. Use the center andradius to write an equation.

Use Three Points to Write an Equation

Solve Graph ΔDEF and construct the perpendicular bisectors of two sides.

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.

Use Three Points to Write an Equation

Check 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.

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.

A. (3, 0)

B. (0, 0)

C. (2, –1)

D. (1, 0)

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.

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.

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 = 32Equation of circle.x2 + (x + 8)2 = 32Substitute x + 8 for y.x2 + x2 + 16x + 64 = 32Evaluate 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.

Use y = x + 8 to find the corresponding y-value.

(–4) + 8 = 4The point of intersection is (–4, 4).

Answer: (–4, 4)

Intersections with Circles

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

A. (2, –2)

B. (2, 2)

C. (–2, –2), (2, 2)

D. (–2, 2), (2, –2)