What is the equation, center, and radius of the circle that passes through (4,3), (0,-5), and (1,2)?

1 Answer
Sep 21, 2016

(x-4)^2+(y+2)^2=25

Radius is 5

Center is (4,-2)

Explanation:

A equation for a circle takes the general form

x^2+y^2+Ax+By+C=0

We are given 3 points that are on the circle so we can write
an equation for each point

4^2+3^2+4A+3B+C=0

0^2+(-5)^2+0A+(-5)B+C=0

1^2+2^2+1A+2B+C=0

Now we simplify where we can

16+9+4A+3B+C=0

25-5B+C=0

5+1A+2B+C=0

Now combine like terms

25+4A+3B+C=0

25-5B+C=0

5+1A+2B+C=0

Now rewrite by subtracting the constant term from both sides.

4A+3B+C=-25

-5B+C=-25

1A+2B+C=-5

I will assume that you know how to solve a system of equations

Doing so you get

A=-8

B=4

C=-5

So our general form for the equation of the circle is

x^2+y^2-8x+4y-5=0

Add 5 to both sides and rewrite

x^2-8x+y^2+4y=5

Now complete the squares as follows

x^2-8x+16+y^2+4y+4=5+16+4

(x-4)^2+(y+2)^2=25