A circle has a center that falls on the line y = 5/6x +8 and passes through (4 ,8 ) and (2 ,5 ). What is the equation of the circle?

1 Answer
Narad T.
May 12, 2017

The equation of the circle is (x-1/3)^2+(y-149/18)^2=4381/324

Explanation:

Let the line L be 6y-5x=48

Let the points A and B be

A=(4,8)

B=(2,5)

The mid-point of AB is

C=((4+2)/2,(8+5)/2)=(3,13/2)

The slope of AB is m=(5-8)/(2-4)=3/2

The slope of the line perpendicular to AB is

m'=-2/3 as mm'=-1

The equation of the line through C and perpendicular to AB is

y-13/2=-2/3(x-3)

6y-39=-4x+12

6y+4x=51 This is line L'

We need the point of intersection of line L and L'

6y-5x=48

6y+4x=51

-9x=-3

x=1/3

6y=5/3+48=149/3

y=149/18

The center of the circle is O=(1/3,149/18)

The radius of the circle is

r^2=((2-1/3)^2+(5-149/18)^2)

=25/9+3481/324

=4381/324

The equation of the circle is

(x-1/3)^2+(y-149/18)^2=4381/324

graph{(y-5/6x-8)((x-1/3)^2+(y-149/18)^2-4381/324)=0 [-8.29, 7.505, 4.45, 12.35]}

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