The locus of all points equidistant from the two points is the perpendicular bisector of the line segment bounded by the two points. The perpendicular bisector goes through the point midway between them, (4,5), and has a slope that is the negative of the reciprocal of the slope of the segment. The segment's slope is 1/5, so the slope of the perpendicular bisector is -5. Its equation is y=-5x+b and b has to satisfy 5=-5(4)+b, so b=25 and the line sought is y=-5x+25.
Part A: That linear function intersects the x-axis when y=0, so 0=-5x+25, and x=5.
Alternatively, the Pythagorean relationship would give (x+1)^2+4^2 = (9-x)^2+6^2, which also solves to x=5.
For part B, since the equation of the perpendicular bisector is -5x+25, if we substitute x=0, we get y=25. Or, (-1)^2+(y-4)^2 = 9^2+(y-6)^2, which also solves to y=25.