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

Question

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

Sadie Allen · Accepted Answer

Any point on the line #color(green)(y=4x-4)#

Consider the vertical line #color(magenta)(x=2)#
Clearly #color(magenta)x=2# goes through the point #color(red)(""(2,6))#
and it intersects #color(blue)(-y+4x=3)# at #color(blue)(""(2,5))#

#color(red)(""(2,6))# is vertically #color(brown)(1)# unit above #color(blue)(""(2,5))#, the intersection point with #color(blue)(-y+4x=3)#

#color(green)(""(2,4))# is vertically #color(brown)1# unit below #color(blue)(""(2,5))#, the intersection point with #color(blue)(-y+4x=3)#

Therefore #color(blue)(-y+4x=3)# bisects the line segment between #color(green)(""(2,4))# and #color(red)(""(2,6))#

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Furthermore (as seen in the image below and considering similar triangles)
any point on a line parallel to #color(blue)(-y+4x=3)# through #color(green)(""(2,4))# will provide, together with #color(red)(""(2,6))# a line segment bisected by #color(blue)(-y+4x=3)#

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#color(blue)(-y+4x=3)# can be written in slope-intercept form as
#color(white)("XXX")color(blue)(y=4x-3)# with a y-intercept at #color(blue)(""(-3))#

If the line through #color(green)(""(2,4))# parallel to #color(blue)(-y+4x=3)# is #color(brown)(1)# unit vertically below #color(blue)(-y+4x=3)#
it will have a y-intercept at #color(green)(""(-4))#
and therefore a slope-intercept form of
#color(white)("XXX")color(green)(y=4x-4)#

Theodore Abad · Answer

undefined The other end of the line segment bisected by the line with the equation $-y + 4x = 3$ can be found by using the midpoint formula. First, we need to determine the coordinates of the midpoint, which lies on the given line.

The midpoint formula is given by:
$$ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $$

Given that one end of the line segment is at (2, 6), we can use the midpoint formula to find the midpoint.

Substitute the given point (2, 6) into the equation $-y + 4x = 3$ to find the y-coordinate of the midpoint:
$$ -y + 4(2) = 3 $$
$$ -y + 8 = 3 $$
$$ -y = -5 $$
$$ y = 5 $$

So, the midpoint is at (2, 5). Now, we can use this midpoint and the given end point (2, 6) to find the other end of the line segment.

Let the coordinates of the other end be (a, b). Using the midpoint formula, we have:
$$ \frac{2 + a}{2} = 2 $$
$$ 2 + a = 4 $$
$$ a = 2 $$

$$ \frac{6 + b}{2} = 5 $$
$$ 6 + b = 10 $$
$$ b = 4 $$

Therefore, the other end of the line segment is at (2, 4).