Equidistant Point on a Line

To find the point on the line 3x + y - 4 = 0 which is equidistant from (-5, 6) and (3, 2), we can use the concept of perpendicular bisector.

1. Finding the Midpoint

The first step is to find the midpoint of the line segment connecting the two given points (-5, 6) and (3, 2). The midpoint formula is given by:

M = [(x1 + x2)/2, (y1 + y2)/2]

Using the given coordinates, we can calculate the midpoint as follows:

M = [(-5 + 3)/2, (6 + 2)/2]
= [-1, 4]

So, the midpoint of the line segment is (-1, 4).

2. Finding the Slope of the Line

The next step is to find the slope of the line passing through the two given points (-5, 6) and (3, 2). The slope formula is given by:

m = (y2 - y1)/(x2 - x1)

Using the given coordinates, we can calculate the slope as follows:

m = (2 - 6)/(3 - (-5))
= -4/8
= -1/2

So, the slope of the line is -1/2.

3. Finding the Perpendicular Slope

Since we are looking for the perpendicular bisector, we need to find the perpendicular slope to the given slope. The perpendicular slope of a line is the negative reciprocal of its slope. Therefore, the perpendicular slope to -1/2 is 2.

4. Finding the Equation of the Perpendicular Bisector

Now that we have the midpoint (-1, 4) and the perpendicular slope 2, we can find the equation of the perpendicular bisector using the point-slope form of a line.

The point-slope form is given by:

y - y1 = m(x - x1)

Substituting the values, we have:

y - 4 = 2(x - (-1))
y - 4 = 2(x + 1)
y - 4 = 2x + 2
y = 2x + 6

So, the equation of the perpendicular bisector is y = 2x + 6.

5. Finding the Intersection Point

To find the point on the line 3x + y - 4 = 0 which is equidistant from (-5, 6) and (3, 2), we need to find the intersection point of the perpendicular bisector (y = 2x + 6) and the given line (3x + y - 4 = 0).

To find the intersection point, we can solve the system of equations:

3x + y - 4 = 0
y = 2x + 6

Substituting the value of y from the second equation into the first equation, we have:

3x + (2x + 6) - 4 = 0
5x + 2 = 0
5x = -2
x = -2/5

Sub

(-1,4)