CHAPTER 12 Circles

OBJECTIVES

Write an equation for a circle. Graph a circle, and identify its center and

radius.

CIRCLE

A circle is the set of points in a plane that are a fixed distance, called the radius, from a fixed point, called the center. Because all of the points on a circle are the same distance from the center of the circle, you can use the Distance Formula to find the equation of a circle.

EXAMPLE 1: USING THE DISTANCE FORMULA TO WRITE THE EQUATION OF A CIRCLE

Write the equation of a circle with center (–3, 4) and radius r = 6.

Solution: Use the Distance Formula with (x2, y2) = (x,

y), (x1, y1) = (–3, 4), and distance equal to the radius, 6.

EXAMPLE

Write the equation of a circle with center (4, 2) and radius r = 7.

Solution:

CIRCLE

Notice that r2 and the center are visible in the equation of a circle. This leads to a general formula for a circle with center (h, k) and radius r.

EXAMPLE 2A: WRITING THE EQUATION OF A CIRCLE

Write the equation of the circle. the circle with center (0, 6) and radius r

= 1 Solution: (x – h)2 + (y – k)2 = r2 (x – 0)2 + (y – 6)2 = 12 x2 + (y – 6)2 = 1

EXAMPLE 2B: WRITING THE EQUATION OF A CIRCLE

Write the equation of the circle. the circle with center (–4, 11) and

containing the point (5, –1) Solution Step 1:Use the distance formula

Step 2: Use the equation of the circle (x + 4)2 + (y – 11)2 = 152

(x + 4)2 + (y – 11)2 = 225

CIRCLES

The location of points in relation to a circle can be described by inequalities. The points inside the circle satisfy the inequality (x – h)2 + (x – k)2 < r2. The points outside the circle satisfy the inequality (x – h)2 + (x – k)2 > r2.

EXAMPLE 3: CONSUMER APPLICATION

Use the map and information given in Example 3 on page 730. Which homes are within 4 miles of a restaurant located at (–1, 1)?

TANGENT

A tangent is a line in the same plane as the circle that intersects the circle at exactly one point. Recall from geometry that a tangent to a circle is perpendicular to the radius at the point of tangency.

EXAMPLE

Write the equation of the line tangent to the circle x2 + y2 = 29 at the point (2, 5).

Solution:

Step 1 Identify the center and radius of the circle.

From the equation x2 + y2 = 29, the circle has center of (0, 0) and radius r = .

Step 2 Find the slope of the radius at the point of tangency and the slope of the tangent.

SOLUTION

Step 3 Find the slope-intercept equation of the tangent by using the point (2, 5) and the slope m = 5/2 .

SOLUTION

The equation of the line that is tangent to x2 + y2 = 29 at (2, 5) is

EXAMPLE

Write the equation of the line that is tangent to the circle 25 = (x – 1)2 + (y + 2)2, at the point (1, –2).

STUDENT GUIDED PRACTICE

Do even problems 2-10 in your book page 826

HOMEWORK

Do 12-20 in your book page 826

VIDEOS

Let’s Watch some videos

CLOSURE

Today we learned about circles Next class we are going to learn about

ellipses