Question

A line segment is bisected by a line with the equation `- y + 3 x = 1`. If one end of the line segment is at `(6 ,3 )`, where is the other end?

Answer 1

`color(blue)((-12/5,29/5)`

Explanation:

We know that the line `-y+3x=1` and the line containing the point `(6.3)` are perpendicular. If two lines are perpendicular then the product of their gradients is `-1`

`-y+3x=1=>y=3x-1 \ \ \ [1]`

This has a gradient of 3. The line containing `(6,3)` therefore has a gradient:

`m*3=-1=>m=-1/3`

Finding the equation of this line using point slope method:

`y-3=-1/3(x-6)=>y=-1/3x+5 \ \ \ \[2]`

Solving `[1] and [2]` simultaneously:

`-1/3x+5-3x+1=0=>x=9/5`

Substitute in `[1]`

`y=3(9/5)-1=22/5`

`(9/5,22/5)` are the coordinates of the midpoint.

Coordinates of the midpoint are found using:

`((x_1+x_2)/2,(y_1+y_2)/2)`

Let the unknown endpoint be `(x_2,y_2)`

Then:

`((6+x_2)/2,(3+y_2)/2)->(9/5,22/5)`

`(6+x_2)/2=9/5=>x_2=-12/5`

`(3+y_2)/2=22/5=>y_2=29/5`

Coordinates of the other endpoint are:

`(-12/5,29/5)`