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

[liz]

Hi, I am 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]

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]

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

 

[HallsofIvy]

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.

 

[ivybond]

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.

If you get stumped I'll be happy to help you along.