If (−2, m) is on a circle with center c(2, 3) and radius 4, what is the value of m?

We know that the equation of a circle is defined as:
x² + y² = r², where the origin is the centre.

Hence, with a centre of (2, 3) and radius of 4 becomes:
(x-2)² + (y-3)² = 4²

Expanding that out, we get:
x² - 4x + 4 + y² - 6y + 9 = 16
x² - 4x + y² - 6y - 3 = 0

Substitute x = -2 as (-2, m) satisfies this equation.

(-2)² - 4(-2) + y² - 6y - 3 = 0
4 + 8 +y² - 6y - 3 = 0
y² - 6y + 9 = 0

Hence, (y - 3)² = 0
y - 3 = 0
And y = 3

So, at x = -2, m = 3