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?
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The point #(-249/85,58/85)#
For a line with a slope that is perpendicular to the given line
#9x + 2y = 3" [1]"#
Swap the coefficients of x and y, change the sign of one of the coefficients, and set it equal to an arbitrary 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 equation of the bisected line is:
#2x - 9y = -12" [2]"#
Multiply equation [1] by 9 and equation [2] by 2:
#81x + 18y = 27" [3]"#
#4x - 18y = -24" [4]"#
Add equation [3] to equation [4]:
#85x = 3#
#x = 3/85#
This is the x coordinate of intersection.
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#
