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

### Some more points…

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.

Once you have answered these, you will be ready to move to the main problem.