Find the equation of the set of all points P(x,y) that is equidistant from (-3,0) and (3,-5).

they lie on the perpendicular bisector of the line between the two points.

That line has slope -5/6, so its perpendicular has slope 6/5

The midpoint of the line segment is (0,-5/2), so the line you want is

y + 5/2 = 6/5 (x-0)

To find the equation of the set of all points equidistant from two points, you need to find the midpoint of the line segment connecting the two points and then find the equation of the perpendicular bisector of that line segment. Let's go through the steps to find the equation.

Step 1: Find the midpoint of the line segment connecting (-3,0) and (3,-5).
The midpoint formula is ( (x1 + x2) / 2 , (y1 + y2) / 2 ).
For this line segment, the midpoint is:
(((-3) + 3) / 2 , (0 + (-5)) / 2 ) = (0, -5/2).

Step 2: Find the slope of the line segment connecting (-3,0) and (3,-5).
The slope formula is (y2 - y1) / (x2 - x1).
Applying it to these two points, we get:
( (-5) - 0 ) / ( 3 - (-3) ) = -5 / 6.

Step 3: Find the negative reciprocal of the slope from Step 2.
The negative reciprocal of a number is obtained by flipping its sign and taking the reciprocal.
So, the negative reciprocal of -5/6 is 6/5.

Step 4: Find the equation of the perpendicular bisector passing through the midpoint.
Using the point-slope form of a line, the equation of the perpendicular bisector passing through (0, -5/2) with a slope of 6/5 can be written as:
y - (-5/2) = (6/5)(x - 0).

Simplifying, we get:
y + 5/2 = 6/5x.

Further simplifying, we can multiply both sides of the equation by 10 to get rid of the fractions:
10y + 25 = 12x.

This equation, 10y + 25 = 12x, represents the set of all points equidistant from (-3,0) and (3,-5).