Question

A line segment is bisected by a line with the equation `3 y + 5 x = 2`. If one end of the line segment is at `( 1 , 4 )`, where is the other end?

Answer 1

The other end of the line is at `(-116/34, 46/34)`

Explanation:

Write the given line in slope-intercept form:

`y = -5/3x + 2/3` [1]

The slope, `m = -5/3`

Because the bisected line is perpendicular, we know that its slope, n, is the negative reciprocal of the bisector:

`n = -1/m`

`n = 3/5`

Use the given point, `(1, 4)` and the slope `3/5` to find the value of b in the slope-intercept form `y = mx + b`:

`4 = 3/5(1) + b`

`b = 4 - 3/5`

`b = 17/5`

The equation of the bisected line is:

`y = 3/5x + 17/5` [2]

Find the x coordinate of the intersection by subtracting equation [1] from equation [2]:

`y - y = 3/5x + 5/3x + 17/5 - 2/3`

`0 =34/15x + 41/15`

`0 =34x + 41`

`-34x = 41`

`x = -41/34`

`Deltax = -41/34 - 1 = -75/34`

The x coordinate of the other end of the line is:

`1 + 2Deltax = 1 + 2(-75/34) = -116/34`

To find the corresponding y coordinate, substitute `-116/34` for x in equation 2:

`y = 3/5(-116/34) + 17/5`

`y = 46/34`