Question

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

Answer 1

The other end is `(179/41,94/41)`

Explanation:

Let's rewrite the equation of the line

`5y-4x=1`

`5y=4x+1`

`y=4/5x+1/5`

The slope is

`m=4/5`

The slope of a line perpendicular is `m'=-5/4`

as `m*m'=-1`

The equation of the segment is

`y-4=-5/4(x-3)`

`y=-5/4x+15/4+4=-5/4x+31/4`

The point of intersection of the lines

`y=4/5x+1/5=-5/4x+31/4`

`(4/5+5/4)x=-1/5+31/4`

`41/20x=161/20`

`x=151/41`

`y=4/5*151/41+1/5`

`y=(604+41)/205`

`y=645/205=129/41`

The point of intersection is `(151/41,129/41)`

Let the other end of the segment be `x_1,y_1`

Then,

`x_1-151/41=151/41-3`

`x_1=151/41+151/41-3=179/41`

and

`y_1-129/41=129/41-4`

`y_1=129/41+129/41-4=94/41`

The other end is `(179/41,94/41)`
graph{(y-4/5x-1/5)(y+5/4x-31/4)((x-3)^2+(y-4)^2-0.01)((x-179/41)^2+(y-94/41)^2-0.01)=0 [-1.985, 9.116, -0.99, 4.56]}