A line segment is bisected by a line with the equation 4 y - 6 x = 8 4y6x=8. If one end of the line segment is at ( 1 , 8 )(1,8), where is the other end?

1 Answer
Jun 13, 2018

color(blue)((67/13,68/13)(6713,6813)

Explanation:

First rearrange 4y-6x=84y6x=8 to the form y=mx+by=mx+b

y=3/2x+2 \ \ \ \[1]

This will be perpendicular to the line through the point (1,8)

We need to find the equation of this line. We know that if two lines are perpendicular then the product of their gradients is bb(-1)

Gradient of [1] is: 3/2

Let bbm be the gradient of the line through (1,8)

Then:

m*3/2=-1=>m=-2/3

Using point slope form of a line and point (1,8):

(y_2-y_1)=m(x_2-x_1)

y-8=-2/3(x-1)

y=-2/3x+26/3 \ \ \ [2]

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

3/2x+2=-2/3x+26/3=>x=40/13

Substitute in [1]

y=3/2(40/13)+2=86/13

Co-ordinates of midpoint (40/13,86/13)

The co-ordinates of the midpoint are given by:

((x_1+x_2)/2,(y_1+y_2)/2)

So:

((x_1+x_2)/2,(y_1+y_2)/2)=(40/13,86/13)

If the end points are (x_1,y_1) and (x_2,y_2)

We have (1,8) and (x_2,y_2)

((1+x_2)/2,(8+y_2)/2)=(40/13,86/13)

Hence:

(1+x_2)/2=40/13=>x_2=67/13

and

(8+y_2)/2=86/13=>y_2=68/13

So other end point is:

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