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

this is a line that is the perpendicular bisector of the line between the given points

To find the equation of the set of all points equidistant from two given points, you can use the concept of midpoint and distance formula.

The midpoint between two points is the average of their x-coordinates and the average of their y-coordinates. Let's find the midpoint between (-3, 0) and (3, -5):

Midpoint x-coordinate: (x₁ + x₂) / 2 = (-3 + 3) / 2 = 0 / 2 = 0
Midpoint y-coordinate: (y₁ + y₂) / 2 = (0 + (-5)) / 2 = -5 / 2 = -2.5

So, the midpoint between the given points is (0, -2.5).

Now, let's use the distance formula to determine the distance between any point P(x, y) and the two given points (-3, 0) and (3, -5).

Distance from P to (-3, 0):
d₁ = √[(x - (-3))² + (y - 0)²] = √[(x + 3)² + y²]

Distance from P to (3, -5):
d₂ = √[(x - 3)² + (y - (-5))²] = √[(x - 3)² + (y + 5)²]

Since the points equidistant from (-3, 0) and (3, -5) are the set of points where d₁ = d₂, we equate the distance formulas:

√[(x + 3)² + y²] = √[(x - 3)² + (y + 5)²]

To eliminate the square roots, we square both sides of the equation:

[(x + 3)² + y²] = [(x - 3)² + (y + 5)²]

Expanding the squared terms:

(x² + 6x + 9 + y²) = (x² - 6x + 9 + y² + 10y + 25)

Canceling out the common terms:

6x + 10y = -6x + 34

Rearranging the equation:

12x + 10y = 34

Therefore, the equation of the set of all points P(x, y) equidistant from (-3, 0) and (3, -5) is 12x + 10y = 34.

To find the equation of the set of all points equidistant from (-3,0) and (3,-5), let's first find the midpoint of the line segment connecting these two points. The midpoint is the point that is equidistant from both endpoints.

The midpoint (M) can be found using the midpoint formula:
M = ((x1 + x2)/2, (y1 + y2)/2)

For (-3,0) and (3,-5), the midpoint (M) is:
M = ((-3 + 3)/2, (0 + -5)/2)
M = (0, -5/2)

Now, let's find the distance from the midpoint (M) to either of the two given points. Let's use (-3,0) for this calculation.

The distance between two points (x1, y1) and (x2, y2) can be found using the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

For M(0, -5/2) and (-3,0), the distance (d) is:
d = sqrt((0 - -3)^2 + (-5/2 - 0)^2)
d = sqrt(3^2 + (-5/2)^2)
d = sqrt(9 + 25/4)
d = sqrt(36/4 + 25/4)
d = sqrt(61/4)

Now, let's set up the equation using the distance between M and (-3,0):
sqrt((x - 0)^2 + (y - (-5/2))^2) = sqrt(61/4)

Simplifying, we get:
(x - 0)^2 + (y + 5/2)^2 = 61/4

Expanding further, we have:
x^2 + (y + 5/2)^2 = 61/4

Therefore, the equation of the set of all points equidistant from (-3,0) and (3,-5) is:
x^2 + (y + 5/2)^2 = 61/4.