Question

A line segment is bisected by a line with the equation `-6 y - x = 3`. If one end of the line segment is at `( 5 , 1 )`, where is the other end?

Answer 1

Coordinates of the other end are `(167/37), -(84/37)`

Explanation:

Assumption : The line bisecting the segment is its perpendicular bisector.

Eqn of line `x + 6y = -3` Eqn (1)
`6y = -x -3`
`y = -(x/6) - (1/2)`
Slope of the line is -(1/6)#

Slope of altitude the line segment is 6
Eqn of line segment is
`y -1 = 6 (x-5)`
`y - 6x = 1-30 = -29` Eqn (2)
Solving Eqns (1) & (2) we get the intersection of the lines or the midpoint of the line segment.
Coordinates of mid point are `(171/37), -(47/37)`

`171/37 = (5+x_2)/2, -(47/37) = (1+y_2)/2` where `x_2, y_2`are the coordinates of the other end point.

`x_2 = (342/37) -5 = 167/37`
`y_2 = -(47/37)-1 = -(84/37)`

Answer 2

`(157/37 , -131/37 )`

Explanation:

First we need to find the equation of the line that is perpendicular to the line `-6y-x=3` and passes through the point `( 5 , 1 )`. Since these lines are perpendicular we can find the gradient of the required line by using the fact that, if `m_1`is the gradient of the known equation, then `m_1*m_2=-1`

`:.`

`-1/6*m_2=-1=>m_2=6`

So second equation is:

`y-1=6(x-5)=>y=6x-29`

The point of intersection of these lines is the coordinates of the midpoint of the line segment.

Solving simultaneously:

`-1/6x-1/2=6x-29=>x=171/37`

`y=6(171/37)-29=-47/37`

We know for a line segment with coordinates `( x_1 , y_2)` and `( 5 , 1 )`, that the coordinates of the midpoint are `((5+x_1)/2 , (1+y_1)/2)`.

Midpoint coordinates:

`(171/37 , -47/37 )`

`:.`

`(5+x_1)/2=171/37=>x_1 =157/37`

`(1+y_1)/2=-47/37=>y_1=-131/37`

Coordinates of end point of line segment:

`(157/37 , -131/37 )`

Plot: