A circle has a center that falls on the line y = 2/3x +7 y=23x+7 and passes through ( 3 ,4 )(3,4) and (6 ,4 )(6,4). What is the equation of the circle?

1 Answer
Jan 18, 2017

(x - 9/2)^2 + (y - 10)^2 = 153/4(x92)2+(y10)2=1534

Explanation:

Circle C: (x - a)^2 + (y - b)^2 = R^2(xa)2+(yb)2=R2

(a,b) in r: b = 2/3 a + 7(a,b)r:b=23a+7

C: (x - a)^2 + (y - 2/3 a - 7)^2 = R^2C:(xa)2+(y23a7)2=R2

(3, 4) in C: (3 - a)^2 + (4 - 2/3 a - 7)^2 = R^2(3,4)C:(3a)2+(423a7)2=R2 // equation 1

(6, 4) in C: (6 - a)^2 + (4 - 2/3 a - 7)^2 = R^2(6,4)C:(6a)2+(423a7)2=R2 // equation 2

Subtract: (1) - (2)

(3 - a)^2 - (6 - a)^2 = 0(3a)2(6a)2=0

3 - a = 6 - a3a=6a   or   3 - a = - (6 - a)3a=(6a)

0a = 30a=3   or   3 + 6 = a + a3+6=a+a

a = 9/2a=92

b = 2/3 * 9/2 + 7 = 10b=2392+7=10

(1) \Rightarrow R^2 = (3 - 9/2)^2 + (4 - 10)^2R2=(392)2+(410)2

R^2 = (3/2)^2 + 36 = (9 + 144)/4 = 153/4R2=(32)2+36=9+1444=1534