A line segment is bisected by a line with the equation # 4 y + 3 x = 8 #. If one end of the line segment is at #( 1 , 8 )#, where is the other end?

Question

A line segment is bisected by a line with the equation # 4  y  + 3  x   =   8  #.  If one end of the line segment is at #(    1  ,  8  )#, where is the other end?

Evelyn Arboleda · Accepted Answer

Over the straight line #4y+3x+19=0# The straight #4y+3x-8 = 3x +4(y-2)=0# passes by point #(0,2)# and has the direction given by the vector #v = (4,-3)#. In parametric form can be written as #p = p_0+lambda v# with #p_0=(0,2)# Given a generic straight point #p#, the symmetrical point to #q = (1,8)# regarding the straight line, is given by #q_S = q +2(p-q) = 2p-q = 2p_0-q+2lambda v# which is a straight line parallel to the initial straight whose equation is given by #q_S=2(0,2)-(1,8)+2\lambda(4,-3)# In Cartesian coordinates we have #x=-1+8\lambda# #y = 4-8-6\lambda# The Cartesian representation gives us #4y+3x+19=0#

Mason Adams · Answer

undefined To find the coordinates of the other end of the line segment, we first need to determine the coordinates of the point where the line segment is bisected by the line $4y + 3x = 8$.

We begin by finding the slope-intercept form of the given line:
$$4y + 3x = 8$$
$$4y = -3x + 8$$
$$y = -\frac{3}{4}x + 2$$

Now, since the line bisects the segment, we need to find the midpoint of the segment. We can use the midpoint formula:
$$M\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

Given one end of the line segment is at (1, 8), let's denote this point as $P(x_1, y_1)$. Now, let's denote the coordinates of the other end of the segment as $Q(x, y)$.

Using the midpoint formula, we can find the coordinates of the midpoint:
$$M\left(\frac{1 + x}{2}, \frac{8 + y}{2}\right)$$

Since the midpoint lies on the line $4y + 3x = 8$, we can substitute the coordinates of the midpoint into the equation of the line and solve for $x$:
$$4\left(\frac{8 + y}{2}\right) + 3\left(\frac{1 + x}{2}\right) = 8$$

Now, solve for $x$ to find the x-coordinate of the other end of the line segment. Once you have the value of $x$, substitute it back into the equation of the line to find the corresponding y-coordinate. This will give you the coordinates of the other end of the line segment.