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Question

Find the relationship between x and y so that the point (x,y) is equidistant from the two points(-2,1) and (2,4). also explain! : )

Answer 1

If you need to find a single point, just use the point halfway between the two. (e.g average the x-coords and the y-coords):

Average of -2, 2 = (2 + -2) / 2 = 0
Average of 1, 4 = (1 + 4) / 2 = 2.5

Midpoint: (0, 2.5)

However, if you need the equation for all points that are equidistant, that would be a line through that point perpendicular to the original line.

For a line perpendicular, it is simply the opposite slope.

The slope of the line through the first two points is given by the equation:
m = (y2-y1) / (x2-x1).
m = ( 4 - 1 ) / (2 - (-2) )
m = 3 / 4

The opposite slope (of a line perpendicular) is the negative reciprocal.
m' = -1 / m
m' = -1 / (3/4)
m' = -4 / 3

So what's a line through (0, 2.5) with a slope of -4/3?

Standard form of a line:
y = mx + b

In this case, b is 2.5, since (0, 2.5) is the y-intercept.

y = (-4/3)x + 2.5

So any points that satisfy this equation will be equidistant from the two points (-2,1) and (2,4)...

Another way to solve this is take a point (x, y). The distance from (-2, 1) to x,y can be figured using the pythagorean theorem:
d² = (x- (-2))² + (y - 1)²
d² = (x + 2)² + (y - 1)²

Similarly from (x, y) to (2, 4):
d² = (x - 2)² + (y - 4)²

Now just equate these and solve for y:
(x + 2)² + (y - 1)² = (x - 2)² + (y - 4)²
(x² + 4x + 4) + (y² -2y + 1) = (x² - 4x +4) + (y² - 8y +16)
x² + 4x + 4 + y² -2y + 1 = x² - 4x +4 + y² - 8y +16

Cancel x² and y² from both sides:
4x - 2y + 5 = -4x - 8y + 20

Group x and y and the numbers together:
6y = -8x + 15

Divide by 6:
y = (-4/3)x + 5/2

This is the same answer as before, so either method will work.

Answer 2

First find the distance of a point (x,y) from the point(-2,1). Now find the distance of the point (x,y) from (2,4). Since the point is equidistant from (x,y) then put them equal and solve it as shown below
(x + 2)^2 + (y - 1)^2 = (x - 2)^2 + (y - 4)^2
x^2 +4x +4 + y^2 -2y + 1 = x^2 - 4x +4 + y^2 -8y + 16
8x +6y -15=0