A point P (2, -1) is equidistant from the points (a, 7) and (-3, a). Find a.

Given point (2, -1) is equidistant from (a, 7) and (-3, a).

i.e. distance between (2, -1) and (a, 7) = distance between (2, -1) and (-3, a)

$\sqrt{(a - 2)^2 + (7 - (-1))^2} = \sqrt{((-3) - 2)^2 + (a - (-1))^2}\\[1em] ⇒ \sqrt{(a - 2)^2 + (7 + 1)^2} = \sqrt{(-3 - 2)^2 + (a + 1)^2}\\[1em] ⇒ \sqrt{(a - 2)^2 + (8)^2} = \sqrt{(-5)^2 + (a + 1)^2}\\[1em] ⇒ (a - 2)^2 + (8)^2 = (-5)^2 + (a + 1)^2\\[1em] ⇒ a^2 + 4 - 4a + 64 = 25 + a^2 + 1 + 2a\\[1em] ⇒ a^2 - 4a + 68 = a^2 + 2a + 26\\[1em] ⇒ - 4a + 68 = 2a + 26\\[1em] ⇒ 68 - 26 = 2a + 4a\\[1em] ⇒ 6a = 42\\[1em] ⇒ a = \dfrac{42}{6}\\[1em] ⇒ a = 7$

Hence, the value of a = 7.