Find a point on the y-axis which is equidistant from the points (5, 2) and (-4, 3).

Let the required point on the y-axis be (0, y).

Given (0, y) is equidistant from (5, 2) and (-4, 3).

i.e. distance between (0, y) and (5, 2) = distance between (0, y) and (-4, 3)

$\sqrt{(5 - 0)^2 + (2 - y)^2} = \sqrt{((-4) - 0)^2 + (3 - y)^2}\\[1em] ⇒ \sqrt{(5)^2 + (2 - y)^2} = \sqrt{(-4)^2 + (3 - y)^2}\\[1em] ⇒ (5)^2 + (2 - y)^2 = (-4)^2 + (3 - y)^2\\[1em] ⇒ 25 + 4 + y^2 - 4y = 16 + 9 + y^2 - 6y\\[1em] ⇒ 29 + y^2 - 4y = 25 + y^2 - 6y\\[1em] ⇒ 29 - 4y = 25 - 6y\\[1em] ⇒ - 4y + 6y = 25 - 29\\[1em] ⇒ 2y = - 4\\[1em] ⇒ y = - \dfrac{4}{2}\\[1em] ⇒ y = - 2$

Hence, the point on the y-axis which is equidistant from the points (5, 2) and (-4, 3) is (0, -2).