A circle has a center that falls on the line #y = 2/9x +8 # and passes through # ( 3 ,1 )# and #(5 ,7 )#. What is the equation of the circle?

1 Answer
Jan 4, 2018

The equation of circle is #(x + 24/5)^2 + (y – 104/15)^2 = 9.8^2#

Explanation:

The center-radius form of the circle equation is

#(x – h)^2 + (y – k)^2 = r^2#, with the center being at the point

#(h, k)# and the radius being #r#. The points #(3,1) and (5,7)#

are on the circle.#:. (3 – h)^2 + (1 – k)^2 =(5 – h)^2 + (7 – k)^2 #

#:. 9 – 6h+cancelh^2 + 1 – 2k+cancelk^2 =25 – 10h+cancelh^2 +49 – 14k+cancelk^2 #

#:. 9 – 6h + 1 – 2k =25 – 10h +49 – 14k # or

# 10h – 6h + 14k – 2k+ =25+49-9 – 1 # or

#4h+12k=64 or h+3k=16 (1); (h,k)# lies on the straight line

#y=2/9x+8 :. k = 2/9h+8 or 9k-2h=72 (2)# . Multiplying

the equation (1) by #2# we get # 2h+6k=32 (3)# . Adding

equation (2) and equation (3) we get #15k=104 or k=104/15#

Putting #k=104/15# in equation (1) we get #h=16-3*104/15# or

#h=16-104/5 or h = (80-104)/5=-24/5 :.# Centre is at

#(h,k) or (-24/5 , 104/15) :. r^2= (3 + 24/5)^2 + (1 – 104/15)^2 #

or #r^2~~96.04 or r~~ 9.8 #. Therefore the equation of circle is

#(x + 24/5)^2 + (y – 104/15)^2 = 9.8^2# [Ans]