Equations of Circles: Translating and Dilating

The equation of a circle with radius r and center ℎ, 𝑘 is

𝑥 − ℎ 2 + 𝑦 − 𝑘 2 = 𝑟2 To translate a circle on the coordinate plane, identify the center of the circle, translate the center, and write the equation for a circle with the same radius at the new center. Translating a circle does not affect the radius of the circle.

Geometry Lesson 95

The equation of a circle is 𝑥 − 3 2 +𝑦 + 2 2 = 25. Translate the circle 4 units

to the left and 2 units down. Write the equation of the translated circle and sketch both circles on the coordinate plane. SOLUTION

The center of the first circle is 3,−2 , and the length of its radius is 5. Sketch the circle. Translating the center four units to the left and 2 units down means the new center is at −1,−4 . The length of the radius is still 5. Therefore, the equation of the new circle is

𝑥 + 1 2 + 𝑦 + 4 2 = 25

Sketch the new circle. As you can see, every point on the new circle is 4 units left and 2 units down from the corresponding point on the old circle.

Geometry Lesson 95

The process for dilating a circle is identical to the process for dilating any other figure on the coordinate plane. It is not practical, however, to apply a dilation to every point on a circle.

Instead, apply the dilation to the radius and the center of the circle only.

Geometry Lesson 95

The equation of a circle is 𝑥2 + 𝑦2 = 49. Apply a dilation centered at the origin with a scale factor of 2. Write the new equation and sketch both circles on the coordinate plane. SOLUTION Since this circle is centered at (0, 0), its center will not change. The radius, however, will become twice as long. Currently, the length of the radius is 7. After the dilation, it will be 14. Therefore, the new equation is 𝑥2 + 𝑦2 = 196.

Geometry Lesson 95

The equation of a circle is 𝑥 − 9 2 +𝑦 + 6 2 = 9. Apply a dilation centered at

the origin with a scale factor of 1

3. Write

an equation for the new circle and sketch both circles. SOLUTION The length of the radius of the original circle is 3, and it is centered at 9,−6 . First, multiply the radius by the scale factor. The radius of the new circle will be 1 unit long. Next, apply the scale factor to each coordinate in the circle’s center.

1

3∙ 9,

1

3∙ −6 = 3,−2

The new center of the circle is (3, -2). Therefore, the equation of the dilated circle is 𝑥 − 3 2 + 𝑦 + 2 2 = 1. Sketch both circles on the coordinate plane.

Geometry Lesson 95

A coordinate plane is overlaid onto the map of a city to help place a tornado-warning siren. The current location of the siren is shown on the coordinate plane, where each unit represents one mile. At its current volume, it is audible to everyone in a 2 mile radius. Suggest a translation and an approximate adjustment to the siren’s volume that would make the siren audible to everyone in the city.

SOLUTION

It makes sense that the siren should be in the center of the city.

The city is bounded by the points −3,4 , −3,−1 , 2,−1 , and 2,4 .

Use the midpoint formula to find the center of the city.

−3 + 2

2,4 − 1

2

−0.5,1.5

The center of the city is at (-0.5, 1.5).

This is a translation of 1.5 units right and 1.5 units down.

The siren at its current strength will not be heard at the

north end of town.

Amplifying the siren by a factor of 3 would make it extend

far beyond the bounds of the city.

Increasing its volume by a factor of 2 will approximately

cover the city, so this is a good estimate.

Geometry Lesson 95

a. The equation of a circle is 𝑥 + 2 2 + 𝑦 − 1 2 = 16

Write the equation of this circle after it is shifted 6 units to the right. Draw the original circle and the shifted circle on the coordinate plane.

Geometry Lesson 95

b. The equation of a circle is 𝑥2 + 𝑦2 = 25. Apply a dilation centered at the origin with a scale factor of 0.5. Write the equation for and sketch the dilated circle.

Geometry Lesson 95

c. The equation of a circle is

𝑥 − 2 2 + 𝑦 − 1 2 = 1. Find the equation of the circle after it is dilated by a scale factor of 4. Draw the original circle and the dilated circle on the coordinate plane.

Geometry Lesson 95

d. June lives 2 miles north and 2 miles west of a tornado siren that can be heard for 3 miles in any direction. The city plans to move the siren 2 miles south but increase its volume by a factor of 1.5. Will the siren still be audible at June’s house?

Geometry Lesson 95

Page 620

Lesson Practice (Ask Mr. Heintz)

Page 621

Practice 1-30 (Do the starred ones first)

Geometry Lesson 95