Question

A line segment is bisected by a line with the equation `- 6 y + 5 x = 4`. If one end of the line segment is at `( 2 , 5 )`, where is the other end?

Answer 1

coordinates of the other end point `color (purple)(362/61, 17/61)`

Explanation:

Assumption : Bisecting line is a perpendicular bisector

Standard form of equation `y=mx +c`
Slope of perpendicular bisector m is given by
`-6y + 5x= 4`
`y= (5/6)x - (2/3)`
`m = (5/6)`

Slope of line segment is
`y - 5 = -(1/m)(x-2)`
`y - 5= (-6/5)(x - 2)`
`5y - 25 = -6x + 12`

`5y + 6x = 37 color (white)((aaaa)` Eqn (1)
`-6y + 5x = 4 color (white)((aaaa)` Eqn (2)

Solving Eqns (1) & (2),

`x= color (green)(242/61)`

`y= color (green)(161/61)`

Mid point `color(green)(242/61, 161/61)`

Let (x1,y1) the other end point.
`(2+x1)/2= 242/37`
`x1= (484/61) - 2 = color(red)( 362/61)`
`(5+y1)/2= 161/61`
`y1=(322/61) - 5 color(red)(17/61)`

Coordinates of other end point `color(red)(362/61, 17/61)`