The center of a circle is (α + 2, α - 5). Find the value of α, given that the circle passes through points (2, -2) and (8, -2).

Given O(α + 2, α - 5) is the center of the circle. A and B are the points on the circle. So, we can say OA = OB = radius.

From distance formula we get,

$OA = \sqrt{(2 - (α + 2))^2 + (-2 - (α - 5))^2} \\[1em] = \sqrt{(2 - α - 2)^2 + (-2 - α + 5)^2} \\[1em] = \sqrt{(-α)^2 + (-α + 3)^2} \\[1em] = \sqrt{α^2 + α^2 + 9 - 6α} \\[1em] = \sqrt{2α^2 - 6α + 9} \qquad \text{….[Eq 1]} \\[1em] OB = \sqrt{(8 - (α + 2))^2 + (-2 - (α - 5))^2} \\[1em] = \sqrt{(8 - α - 2)^2 + (-2 - α + 5)^2} \\[1em] = \sqrt{(6 - α)^2 + (3 - α)^2} \\[1em] = \sqrt{36 + α^2 - 12α + 9 + α^2 - 6α} \\[1em] = \sqrt{2α^2 + 45 - 18α} \qquad \text{….[Eq 2]}$

Comparing both the Equation, since they are equal to radius,

$\Rightarrow \sqrt{2α^2 - 6α + 9} = \sqrt{2α^2 + 45 - 18α} \\[1em]$

Squaring both sides we get,

$\Rightarrow 2α^2 - 6α + 9 = 2α^2 + 45 - 18α \\[1em] \Rightarrow 2α^2 - 2α^2 - 6α + 18α + 9 - 45 = 0 \\[1em] \Rightarrow 12α - 36 = 0 \\[1em] \Rightarrow 12α = 36 \\[1em] \Rightarrow α = 3.$

Hence, the value of α = 3.