# Finding the equation of a circle from given point on the graph

Hi, im sure this question is really easy, but i don't have a clue how to do this question....

Three points are P (-2, 7), Q (2,3), and R (4, 5). Find the equation of the circle which passes through points P, G, and R.

Thanks lots to anyone who helps. liz x

Galileo
Homework Helper
Well, if you could find the center of the circle the rest is easy.
Suppose you draw a circle and pick two arbitrary (different) points on that circle. What construction with only these points would allow you to draw a line that passes through the center of the circle?
Once you know this, you can apply it to 2 pairs of points from P,Q,R and construct the center.

TD
Homework Helper
Or, an alternative method, you can consider the triangle formed by the 3 points. Then try to find the circumcircle.

HallsofIvy
Homework Helper
Those are very good geometric ways to specify the circle. Since the original question asked for the equation, you could also do this: the equation of a general circle with center (a,b) and radius R is
(x- a)2+ (y- b)2= R2.

Since the circle passes through (-2, 7), x=-2, y= 7 must satisfy that:
(-2-a)2+ (7-b)2= R2.
Do the same with the other two points and you have 3 equations to solve for the 3 unknowns a, b, and R.

liz said:
Three points are P (-2, 7), Q (2,3), and R (4, 5). Find the equation of the circle which passes through points P, G, and R.

It always helps plot given points because the question might present a special case.

It also helps a lot use graph paper.

After you plot these points you might notice that PQ and QR go through the grid intersictions in a special way.
Can you prove what you see? There is a certain rule you should apply.

Since a triangle PQR is special, there is an interesting property of the center of its circumcircle. It makes it a snap to find its coordinates and radius, and equation of the circle.