College Algebra 9th txtbk

SECTION 2–2 Distance in the Plane 129
EXAMPLE 7 Finding the Center and Radius of a Circle
Find the center and radius of the circle with equation x 2 y 2 6x 4y 23.
SOLUTION
We transform the equation into the form (x h) 2 (y k) 2 r 2 by completing the square
relative to x and relative to y (see Section 1-5). From this standard form we can determine
the center and radius.
(x 2 6x ) ( y 2 4y ) 23
(x 2 6x 9) ( y 2 4y 4) 23 9 4
Center:
Radius:
x 2 y 2 6x 4y 23
(x 3) 2 ( y 2) 2 36
[ x (3)] 2 ( y 2) 2 6 2
(h, k) (3, 2)
r 136 6
Group together the terms involving x
and those involving y.
Complete the squares.
Factor each trinomial.
Write 3 as (3) to identify h.
MATCHED PROBLEM 7 Find the center and radius of the circle with equation x 2 y 2 8x 10y 25.
ANSWERS TO MATCHED PROBLEMS
1. 13
2. 1173
3. M ( 1 2, 2) (0.5, 2); d(A, B) 185; d(A, M ) 121.25 d(M, B) 1 2 d(A, B)
5
y
5
A (4, 1)
5
x
5
B (3, 5)
4. B (3, 3)
5. x 2 + y 2 9 6. (x 3) 2 ( y 2) 2 9
y
y
5
(0, 3)
(3, 1)
(3, 0)
5
(3, 0)
5
x
(0, 2)
5
C
(3, 2)
x
(6, 2)
5
(0, 3)
5
(3, 5)
7. (x 4) 2 ( y 5) 2 16; radius: 4, center: (4, 5)