Find the equation in standard form of the circle with points (2,3), (3,4), (-1,2)

Question

I can't keep up with the solution. I need it for my quiz on tuesday. 

Answer 1

The standard form for the equation of a circle: (x - h)² + (y - k)² = r²

 

The three points will have the same distance - radius - from the center of the circle, (h, k)

 

The distance from (2, 3) to (h, k) will equal the distance from (3, 4) to (h, k) will equal the distance from (-1, 2) to (h, k).

 

Use the distance formula d = √((x - h)² + (y - k)²) three times; once for each given point. The three distances will be equal if there are no algebra mistakes.

 

Then center is (-1, 7) and the radius length is found to be 5.

 

A²

 

Answer 2

center at (X,Y)

 

(x-2)^2 + (y-3)^2  =  (x-3)^2 + (y-4)^2  

 

x^2 - 4x + 4  + y^2 - 6y + 9  = x^2 - 6x + 9 + y^2 - 8y + 16

 

-4x + 4 - 6y + 9 = -6x + 9 - 8y + 16

 

2x + 2y -12 = 0

 

x + y  = 6

 

 

(x-2)^2 + (y-3)^2 = (x+1)^2 + (y-2)^2

 

 

x^2 - 4x + 4 + y^2 - 6y + 9 = x^2 + 2x + 1 + y^2 - 4y + 4

 

-4x + 4 - 6y + 9 = 2x + 1 - 4y + 4

 

8 = 6x + 2y 

 

4 = 3x + y

 

 

4 = 3(6-y) + y

 

4 = 18 - 3y + y

 

4 = 18 - 2y

 

-14 = -2y

 

y = 7

 

x=-1

 

 

Using the following to check:

(x-3)^2 + (y-4)^2 = (x+1)^2 + (y-2)^2

 

(-1-3)^2 + (7-4)^2 = (-1+1)^2 + (7-2)^2

 

(-4)^2 + 3^2 = 0^2 + 5^2

 

16 + 9 = 0 + 25

 

25 = 25

 

 

YES IT WORKS

 

So the radius us sqrt(25)=5

 

and the center is at (-1, 7)

 

 

The equation is

(x+1)^2 + (y-7)^2 = 25