A and B are the points $(-7, -3)$ and $(1, 5)$ - Scottish Highers Maths - Question 4 - 2016

The radius of the circle is half the length of the diameter AB.
The length of segment AB can be calculated using the distance formula:

$$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

where $(x_1, y_1) = (-7, -3)$ and $(x_2, y_2) = (1, 5)$.
Calculating the distance:

$$D = \sqrt{(1 - (-7))^2 + (5 - (-3))^2} = \sqrt{(1 + 7)^2 + (5 + 3)^2} = \sqrt{8^2 + 8^2} = \sqrt{64 + 64} = \sqrt{128} = 8\sqrt{2}$$

Thus, the radius $r$ is:

$$r = \frac{D}{2} = \frac{8\sqrt{2}}{2} = 4\sqrt{2}$$

The standard form of the equation of a circle with center $(h, k)$ and radius $r$ is given by:

$$(x - h)^2 + (y - k)^2 = r^2$$

Substituting the values $(h, k) = (-3, 1)$ and $r = 4\sqrt{2}$:

$$(x + 3)^2 + (y - 1)^2 = (4\sqrt{2})^2$$

Calculating $r^2$:

$$(4\sqrt{2})^2 = 16 \cdot 2 = 32$$

Thus, the equation of the circle is:

$$(x + 3)^2 + (y - 1)^2 = 32$$