A line segment is bisected by a line with the equation # 2 y + 9 x = 3 #. If one end of the line segment is at #(3 ,2 )#, where is the other end?

Question

A line segment is bisected by a line with the equation #     2 y  + 9   x   =   3  #.  If one end of the line segment is at #(3  ,2  )#, where is the other end?

Autumn Armstrong · Accepted Answer

The point #(-249/85,58/85)#

Regarding a line whose slope is perpendicular to the specified line

#9x + 2y = 3" [1]"#

Exchange the x and y coefficients, alter one of the coefficients' signs, and set it equal to any random constant:

#2x - 9y = C#

To find the value of the constant, substitute the point #(3,2)# into the equation:

#2(3) - 9(2) = C#

#C = -12#

The bisected line's equation is as follows:

#2x - 9y = -12" [2]"#

Equation [1] should be multiplied by 9 and equation [2] by 2:

#81x + 18y = 27" [3]"# #4x - 18y = -24" [4]"#

Combine equations [3] and [4] as follows:

#85x = 3#

#x = 3/85#

This is the intersection's x coordinate.

Let #Deltax = # the change from the original x coordinate, 3, to the x coordinate of intersection:

#Deltax = (3/85 - 3) = -252/85#

Let #x_1 = # the x coordinate of the other end of the line segment.

#x_1 = 2Deltax + 3#

#x_1 = -504/85 + 3#

#x_1 = -249/85#

To find the corresponding y coordinate, substitute the value of #x_1# into equation [2]:

#2(-249/85) - 9y_1 = -12#

#y_1 = 58/85#

Noah Atkinson · Answer

undefined To find the other end of the line segment bisected by the line $2y + 9x = 3$, given that one end is at (3, 2), we can use the midpoint formula.

First, let's find the midpoint of the line segment. The midpoint formula is given by:
$$ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $$

Given one endpoint at (3, 2), let's denote the coordinates of the other endpoint as (x, y). We know that the midpoint is on the line $2y + 9x = 3$, so we can use this information to find the other endpoint.

Using the midpoint formula and substituting the midpoint into the equation of the line, we get:
$$ \frac{3 + x}{2} = \frac{2 + y}{2} $$
$$ 3 + x = 2 + y $$
$$ y = x + 1 $$

Now, substitute $ y = x + 1 $ into the equation of the line $2y + 9x = 3$ to solve for x:
$$ 2(x + 1) + 9x = 3 $$
$$ 2x + 2 + 9x = 3 $$
$$ 11x = 1 $$
$$ x = \frac{1}{11} $$

Now that we have the x-coordinate, we can find the y-coordinate using $ y = x + 1 $:
$$ y = \frac{1}{11} + 1 $$
$$ y = \frac{12}{11} $$

Therefore, the other end of the line segment is at the point $ \left( \frac{1}{11}, \frac{12}{11} \right) $.