The fact that the circle is TANGENT to the y axis means that the radius from the center of the circle, (5, -2), to that point is perpendicular to the y-axis. And since the y-axis is vertical, that radius is horizontal. That means the circle is tangent to the y-axis at (0, -2). And that, in turn, means that the radius of the circle is 5.
The point (-4, 3) will lie on that circle if and only if the distance from (5, -2) to (-4, 3) is 5.