X and y are the points (3,-2) and (4,6), produce the line xy in both direction and find the coordinates of A and B such that AX = XY = BY?

To find the equation of the line passing through points X(3,-2) and Y(4,6), we first need to find the slope of the line.

Slope (m) = (y2 - y1) / (x2 - x1)
m = (6 - (-2)) / (4 - 3)
m = 8

Now that we have the slope, we can use the point-slope form of the equation of a line to find the equation of the line passing through points X and Y.

y - y1 = m(x - x1)
y - (-2) = 8(x - 3)
y + 2 = 8x - 24
y = 8x - 26

So, the equation of the line passing through points X and Y is y = 8x - 26.

To find points A and B such that AX = XY = BY, we need to find the midpoint of line XY, which is also the point A and B.

Midpoint formula:
x = (x1 + x2) / 2
y = (y1 + y2) / 2

For points X(3,-2) and Y(4,6):
x = (3 + 4) / 2 = 3.5
y = (-2 + 6) / 2 = 2

Therefore, the coordinates of points A and B are A(3.5, 2) and B(3.5, 2).

Line xy:

The equation of the line passing through points X(3,-2) and Y(4,6) is:

```
y - (-2) = (6 - (-2)) / (4 - 3) - (x - 3)
y + 2 = 8 / 1 - (x - 3)
y = 8x - 26
```

Line in the opposite direction:

The line in the opposite direction has the same slope but a different y-intercept. The equation of this line is:

```
y - 6 = -(8 / 1) - (x - 4)
y = -8x + 42
```

Coordinates of A and B:

To find the coordinates of A and B, we need to find where the two lines intersect. Solving the system of equations:

```
y = 8x - 26
y = -8x + 42
```

gives us the point of intersection:

```
x = 3, y = 4
```

So, A = (3,4) and B = (3,4).

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