# A line segment is bisected by a line with the equation  4 y - 6 x = 8 . If one end of the line segment is at ( 1 , 8 ), where is the other end?

Jun 13, 2018

color(blue)((67/13,68/13)

#### Explanation:

First rearrange $4 y - 6 x = 8$ to the form $y = m x + b$

$y = \frac{3}{2} x + 2 \setminus \setminus \setminus \setminus \left[1\right]$

This will be perpendicular to the line through the point $\left(1 , 8\right)$

We need to find the equation of this line. We know that if two lines are perpendicular then the product of their gradients is $\boldsymbol{- 1}$

Gradient of $\left[1\right]$ is: $\frac{3}{2}$

Let $\boldsymbol{m}$ be the gradient of the line through $\left(1 , 8\right)$

Then:

$m \cdot \frac{3}{2} = - 1 \implies m = - \frac{2}{3}$

Using point slope form of a line and point $\left(1 , 8\right)$:

$\left({y}_{2} - {y}_{1}\right) = m \left({x}_{2} - {x}_{1}\right)$

$y - 8 = - \frac{2}{3} \left(x - 1\right)$

$y = - \frac{2}{3} x + \frac{26}{3} \setminus \setminus \setminus \left[2\right]$

The intersection of $\left[1\right]$ and $\left[2\right]$ will be the midpoint of the line segment. Solving these simultaneously:

$\frac{3}{2} x + 2 = - \frac{2}{3} x + \frac{26}{3} \implies x = \frac{40}{13}$

Substitute in $\left[1\right]$

$y = \frac{3}{2} \left(\frac{40}{13}\right) + 2 = \frac{86}{13}$

Co-ordinates of midpoint $\left(\frac{40}{13} , \frac{86}{13}\right)$

The co-ordinates of the midpoint are given by:

$\left(\frac{{x}_{1} + {x}_{2}}{2} , \frac{{y}_{1} + {y}_{2}}{2}\right)$

So:

$\left(\frac{{x}_{1} + {x}_{2}}{2} , \frac{{y}_{1} + {y}_{2}}{2}\right) = \left(\frac{40}{13} , \frac{86}{13}\right)$

If the end points are $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$

We have $\left(1 , 8\right)$ and $\left({x}_{2} , {y}_{2}\right)$

$\left(\frac{1 + {x}_{2}}{2} , \frac{8 + {y}_{2}}{2}\right) = \left(\frac{40}{13} , \frac{86}{13}\right)$

Hence:

$\frac{1 + {x}_{2}}{2} = \frac{40}{13} \implies {x}_{2} = \frac{67}{13}$

and

$\frac{8 + {y}_{2}}{2} = \frac{86}{13} \implies {y}_{2} = \frac{68}{13}$

So other end point is:

color(blue)((67/13,68/13)