Analytical Geometry : Circle Equations

Circle Equations from Math is Fun 

How to Find Equation of a Circle Passing 3 Given Points 

7 methods included ; Amazing !!

Practice finding the equation of a Circle given 3 points -- 

Q #1 : (1, 3), (7, 3) and (1, -3)

Answer : (x -4)² + y² = 18

Q #2  : (3, 4), (3, -4), (0, 5)

Answer : x² + y² = 25

Q #3 : A (1, 1), B (2, 4), C (5, 3)

Answer : (x-3)² + (y -2)² = 5

Solution : 

The midpoint of line AB on the Cartesian plane is $(\frac{3}{2},\frac{5}{2})$ and the slope is $(\frac{3}{1})$ so the slope of the perpendicular bisector of line AB is $(\frac{-1}{3})$.

The equation of the line bisect line AB and perpendicular to line AB is thus :  

y - $(\frac{5}{2}$) =$\frac{-1}{3}$ [x - $(\frac{3}{2})$] --- equation 1

The midpoint of line BC on the Cartesian plane is $(\frac{7}{2},\frac{7}{2})$ and the slope is $(\frac{-1}{3})$ so the slope of the perpendicular bisector of line BC is 3.

And the equation of the line bisect line BC and perpendicular to line BC is 

y - $(\frac{7}{2})$ = 3 [x - $(\frac{7}{2})$] --- equation 2

Solve the two equations for x and y and you have the center of the circle being (3, 2)

Use distance formula from the center circle to any point to get the radius = 

$\sqrt{5}$.

The answer is : (x - 3)² + (x - 2)² = 5

More practices on similar questions :  (Answers below for self check)

Q #1 : A (2, 5) , B (2, 13) ,  and C (-6, 5 )

Q #2 : A (0, 7), B ( 6, 5 ), and C (-6, -11 )

Q #3 : A (3, -5) , B (-4, 2) and C (1, 7 )








Answer key :

#1 :  (x - 2) ² + ( y - 9 ) ² = 32

#2 :   x² + (y + 3)² = 100

#3 :  (x - 2) ² + ( y -1) ² = 37