Question

A line segment is bisected by a line with the equation `-6 y + 9 x = 2`. If one end of the line segment is at `( 5 , 1 )`, where is the other end?

Answer 1

The point is `(-9/13, 187/39)`

Explanation:

Write the given equation in slope-intercept form:

`y = 3/2x - 1/3`

The slope is `3/2` any line perpendicular will have `-2/3` slope.

Use the point-slope form of the equation of a line, to find the equation of the bisected line:

`y - 1 = -2/3(x - 5)`

`y = -2/3x + 13/3`

Subtract the second line from the first:

`0 = (3/2 + 2/3)x - 14/3`

`13/6x = 14/3`

`x = 28/13`

The change in x from `5` to `28/13` is:

`28/13 - 13/13(5) = -37/13`

Add twice that number to 5 to get the x coordinate of the other end of the line segment:

13/13(5) - 74/13 = -9/13

Substitute into the equation of the line for the y coordinate:

`y = -2/3(-9/13) + 13/3`

`y = 18/39 + 169/39`

`y = 187/39`