Math Help - Find the equation of the line CB

1. Find the equation of the line CB

hello, this is my question:

The point A(3,2) is one end of the diameter AB of a circle. Both A and B lie on the line y=1/3x+1.
The point C(1,4) lies on the circumference of the circle.

Fine the equation of CB.

im not too sure at all how i would do this question so i would require some help please! could you write up step by step what i have to do and explain fully so i can try and understand what is going on?!

thanks!

2. Re: Find the equation of the line CB

Originally Posted by andyboy179
.....could you write up step by step what i have to do and
explain fully so i can try and understand what is going on?!

3. Re: Find the equation of the line CB

Originally Posted by andyboy179
hello, this is my question:

The point A(3,2) is one end of the diameter AB of a circle. Both A and B lie on the line y=1/3x+1.
The point C(1,4) lies on the circumference of the circle.

Fine the equation of CB.

im not too sure at all how i would do this question so i would require some help please! could you write up step by step what i have to do and explain fully so i can try and understand what is going on?!

thanks!
Let's denote center of circle as :

$O(x_O,y_O)$

step 1 .

Solve following system of equations in order to find center of circle :

$\begin{cases}(x_O-x_A)^2+(y_O-y_A)^2=(x_O-x_C)^2+(y_O-y_C)^2 \\y_O=\frac{1}{3}x_O+1\end{cases}$

step 2 .

calculate point B :

$x_B=2x_O-x_A$

$y_B=2y_O-y_A$

step 3 .

Write equation of the line CB using equation of the line through two points .

$y-y_C=\frac{y_C-y_B}{x_C-x_B}(x-x_C)$

4. Re: Find the equation of the line CB

Hi andyboy79,
The following may be of more help
1 the perpendicular bisector of AC passes thru the circle center
2 solution of y=1/3x+1 and ! above gives the center
3 radius of circle = distance between A and center
4 determine point B using radius and y=1/3x+1
5 CB has same slope as 1.Use point slope formular to get equation of CBl