# A line segment is bisected by line with the equation  6 y - 7 x = 3 . If one end of the line segment is at (4 ,2 ), where is the other end?

May 16, 2018

color(blue)((74/85,398/85)

#### Explanation:

First we note that if two lines are perpendicular then the product of their gradients is $- 1$

We need to find the equation of a line that contains the point $\left(4 , 2\right)$ and is perpendicular to $6 y - 7 x = 3$

Rearranging $\setminus \setminus \setminus \setminus 6 y - 7 x = 3$

$y = \frac{7}{6} x + \frac{1}{2} \setminus \setminus \setminus \setminus \left[1\right]$

If the gradient for our given line be $m$, then:

$m \cdot \frac{7}{6} = - 1 \implies m = - \frac{6}{7}$

Using point slope form of a line:

$y - 2 = - \frac{6}{7} \left(x - 4\right)$

$y = - \frac{6}{7} x + \frac{38}{7} \setminus \setminus \setminus \setminus \left[2\right]$

Finding the point of intersection.

Solve $\left[1\right]$ and $\left[2\right]$ simultaneously:

$- \frac{6}{7} x + \frac{38}{7} - \frac{7}{6} x - \frac{1}{2} = 0$

$x = \frac{207}{85}$

Substituting in $\left[1\right]$

$\frac{7}{6} \left(\frac{207}{85}\right) + \frac{1}{2} = \frac{284}{85}$

$\left(\frac{207}{85} , \frac{284}{85}\right)$ are the coordinates of the midpoint:

The coordinates for the midpoint of a line is given by:

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

Therefore:

$\left(\frac{4 + {x}_{2}}{2} , \frac{2 + {y}_{2}}{2}\right) \to \left(\frac{207}{85} , \frac{284}{85}\right)$

Hence:

$\frac{4 + x}{2} = \frac{207}{85} \implies x = \frac{74}{85}$

$\frac{2 + y}{2} = \frac{284}{85} \implies y = \frac{398}{85}$

Coordinates of the other end are:

$\left(\frac{74}{85} , \frac{398}{85}\right)$

PLOT: 