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?

1 Answer
Somebody N.
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 (4,2) and is perpendicular to 6y-7x=3

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

y=7/6x+1/2 \ \ \ [1]

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

m*7/6=-1=>m=-6/7

Using point slope form of a line:

y-2=-6/7(x-4)

y=-6/7x+38/7 \ \ \ [2]

Finding the point of intersection.

Solve [1] and [2] simultaneously:

-6/7x+38/7-7/6x-1/2=0

x=207/85

Substituting in [1]

7/6(207/85)+1/2=284/85

(207/85,284/85) are the coordinates of the midpoint:

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

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

Therefore:

((4+x_2)/2,(2+y_2)/2)->(207/85,284/85)

Hence:

(4+x)/2=207/85=>x=74/85

(2+y)/2=284/85=>y=398/85

Coordinates of the other end are:

(74/85,398/85)

PLOT:

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