Question

A line segment is bisected by a line with the equation `- 3 y + 5 x = 8`. If one end of the line segment is at `( 7 , 9 )`, where is the other end?

Answer 1

Coordinates of the other end point `color(blue)(-61/17, -147/17)`

Explanation:

Assumption : Bisecting line is a perpendicular bisector

Standard form of equation `y=mx +c`
Slope of perpendicular bisector m is given by
`-3y + 5x= 8`
`y= (5/3)x - (8/3)`
`m= (5/3)`

Slope of line segment is
`y - 9 = -(1/m)(x-7)`
`y - 9= (-3/5)(x - 7)`
`5y - 45 = -3x + 21`

`-3y + 5x = 8 color (white)((aaaa)` Eqn (1)
`5y + 3x = 66 color (white)((aaaa)` Eqn (2)

Solving Eqns (1) & (2),

`x= color (purple)29/17`

`y= color (purple)3/17`
Mid point `color(purple)(29/17, 3/17)`

Let (x1,y1) the other end point.
`(7+x1)/2= 29/17`
`x1= (58/17) - 7 = color(red)( -61/17)`
`(9+y1)/2= 3/17`
`y1=(6/17) - 9 = color(red)(-147/17)`

Coordinates of other end point `color(red )(-61/17, -147/17)`