A line segment is bisected by a line with the equation # - y + 3 x = 1 #. If one end of the line segment is at #(6 ,3 )#, where is the other end?

Question

A line segment is bisected by a line with the equation #   -   y  +  3  x   =   1  #.  If one end of the line segment is at #(6  ,3  )#, where is the other end?

Sadie Allen · Accepted Answer

#color(blue)((-12/5,29/5)#

We know that the line #-y+3x=1# and the line containing the point #(6.3)# are perpendicular. If two lines are perpendicular then the product of their gradients is #-1#

#-y+3x=1=>y=3x-1 \ \ \ [1]#

This has a gradient of 3. The line containing #(6,3)# therefore has a gradient:

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

Using the point slope method, find the equation for this line:

#y-3=-1/3(x-6)=>y=-1/3x+5 \ \ \ \[2]#

Solving #[1] and [2]# simultaneously:

#-1/3x+5-3x+1=0=>x=9/5#

Substitute in #[1]#

#y=3(9/5)-1=22/5#

#(9/5,22/5)# are the coordinates of the midpoint.

The midpoint's coordinates can be found by using:

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

Let the unknown endpoint be #(x_2,y_2)#

Then:

#((6+x_2)/2,(3+y_2)/2)->(9/5,22/5)#

#(6+x_2)/2=9/5=>x_2=-12/5#

#(3+y_2)/2=22/5=>y_2=29/5#

The opposite endpoint's coordinates are:

#(-12/5,29/5)#

Theodore Abad · Answer

undefined To find the other end of the line segment bisected by the line -y + 3x = 1, given that one end is at (6, 3), we first need to find the equation of the line segment represented by the bisecting line. Then, we can find the intersection point of this line with the given line segment to locate the other end.

1. **Find the equation of the bisecting line:**

Given the equation of the bisecting line is -y + 3x = 1, we rearrange it into slope-intercept form (y = mx + b):

-y + 3x = 1
=> y = 3x - 1

So, the slope of the bisecting line is 3.

2. **Find the midpoint of the line segment:**

Given one end of the line segment is at (6, 3), we'll use the midpoint formula to find the midpoint. Let's denote the other end as (a, b), then the midpoint (M) is:

M = ((6 + a)/2, (3 + b)/2)

3. **Use the midpoint and slope to find the other end:**

Since the bisecting line goes through the midpoint (M) and has a slope of 3, we can use the point-slope form to find the equation of this line:

y - y₁ = m(x - x₁)

Substituting the midpoint coordinates (6 + a)/2 and (3 + b)/2 for (x₁, y₁), we have:

y - (3 + b)/2 = 3(x - (6 + a)/2)

Now, we substitute the point (6, 3) into this equation to solve for the other end (a, b).

3 = 3(6 - (6 + a)/2)
3 = 18 - 3(a + 6)/2
3 = 18 - (3/2)(a + 6)
3 = 18 - (3/2)a - 9
9 = (3/2)a
a = 6

Substitute a = 6 into the equation to find b:

b - 3/2 = 3(6 - (6 + 6)/2)
b - 3/2 = 3(6 - 6)
b - 3/2 = 0
b = 3/2

Therefore, the other end of the line segment is at (6, 3/2).