C HAPTER 12 Circles

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

C IRCLE 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.

E XAMPLE 1: U SING THE D ISTANCE F ORMULA TO W RITE THE E QUATION OF A C IRCLE Write the equation of a circle with center (–3, 4) and radius r = 6. Solution: Use the Distance Formula with ( x 2, y 2 ) = ( x, y ), ( x 1, y 1 ) = (–3, 4), and distance equal to the radius, 6.

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

CIRCLE Notice that r 2 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.

E XAMPLE 2A: W RITING THE E QUATION OF A C IRCLE Write the equation of the circle. the circle with center (0, 6) and radius r = 1 Solution: (x – h ) 2 + ( y – k ) 2 = r 2 (x – 0) 2 + ( y – 6) 2 = 1 2 x 2 + ( y – 6) 2 = 1

E XAMPLE 2B: W RITING THE E QUATION OF A C IRCLE 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 = 15 2 ( 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 r 2.

E XAMPLE 3: C ONSUMER A PPLICATION 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)?

T ANGENT 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.

E XAMPLE Write the equation of the line tangent to the circle x 2 + y 2 = 29 at the point (2, 5). Solution: Step 1 Identify the center and radius of the circle. From the equation x 2 + y 2 = 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 x 2 + y 2 = 29 at (2, 5) is

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

S TUDENT G UIDED P RACTICE Do even problems 2-10 in your book page 826

H OMEWORK Do in your book page 826

VIDEOS Let’s Watch some videos

C LOSURE Today we learned about circles Next class we are going to learn about ellipses