The points $A(-1,5)$ and $B(5,3)$ lie on the circumference of a circle. You are given that $AB$ is a diameter of the circle.

1. Find an equation for the circle.

2. Verify that the point $C(1,7)$ lies on the circle, and show that $AC$ is perpendicular to $CB.$

3. Find the area of the triangle $\mathrm{△}ABC.$

Here is a picture of the situation:

If you can find $M$ (the midpoint of $AB$), you know the centre of the circle.

If you can find $|AB|$ (the distance from $A$ to $B$), you know the diameter, and halving this gives you the radius.

Now you have all the information you need to write down the equation of the circle.

Make sure the coordinates of $C$ satisfy the equation. For perpendicularity, consider the gradient of $AC$ and $BC$ - do they multiply to give $-1$?

Since $AC$ is perpendicular to $BC$, you just need to find $\frac{1}{2}|AC||BC|.$