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# Finding Circles

#### Warm-up

#### Some more points...

#### Main problem

**This is an Underground Mathematics resource.**

*Underground Mathematics is hosted by Cambridge Mathematics. The project was originally funded by a grant from the UK Department for Education to provide free web-based resources that support the teaching and learning of post-16 mathematics.*

*Visit the site at undergroundmathematics.org to find more resources, which also offer suggestions, solutions and teacher notes to help with their use in the classroom.*

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Age 16 to 18

Challenge Level

The points $(3,8)$ and $(5,2)$ lie at the ends of a diameter of a circle. What is the equation of the circle?

In the previous question, two points were enough to specify a circle. But is that always the case? For each of the following three statements, is the answer ALWAYS, SOMETIMES or NEVER? Do justify your answers!

You might like to explore your ideas with Geogebra.

- If we specify two (distinct) points in the plane, there is a unique circle that passes through them.
- If we specify three (distinct) points in the plane, there is a unique circle that passes through them.
- If we specify four (distinct) points in the plane, there is a unique circle that passes through them.

We now know that given three distinct points in the plane, there is (usually) a unique circle passing through them.

So if we are given three points, we would like to find the equation of the circle that passes through them. How many ways can you find to do this?

Here are some examples for you to try your ideas out on. For each set of three points, find the equation of the circle passing through them.

It is certainly worth sketching a graph to help you understand what is going on in each case!

Which of your approaches is the most effective?

- $A(3,2)$, $B(3,6)$, $C(5,8)$
- $A(0,4)$, $B(4,0)$, $C(-6,0)$
- $A(-1,-5)$, $B(-2,2)$, $C(2,-1)$
- $A(3,3)$, $B(1,-2)$, $C(4,1)$