A line segment is bisected by a line with the equation # 5 y -4 x = 1 #. If one end of the line segment is at #(3 ,4 )#, where is the other end?

1 Answer
Narad T.
May 11, 2017

The other end is #(179/41,94/41)#

Explanation:

Let's rewrite the equation of the line

#5y-4x=1#

#5y=4x+1#

#y=4/5x+1/5#

The slope is

#m=4/5#

The slope of a line perpendicular is #m'=-5/4#

as #m*m'=-1#

The equation of the segment is

#y-4=-5/4(x-3)#

#y=-5/4x+15/4+4=-5/4x+31/4#

The point of intersection of the lines

#y=4/5x+1/5=-5/4x+31/4#

#(4/5+5/4)x=-1/5+31/4#

#41/20x=161/20#

#x=151/41#

#y=4/5*151/41+1/5#

#y=(604+41)/205#

#y=645/205=129/41#

The point of intersection is #(151/41,129/41)#

Let the other end of the segment be #x_1,y_1#

Then,

#x_1-151/41=151/41-3#

#x_1=151/41+151/41-3=179/41#

and

#y_1-129/41=129/41-4#

#y_1=129/41+129/41-4=94/41#

The other end is #(179/41,94/41)#
graph{(y-4/5x-1/5)(y+5/4x-31/4)((x-3)^2+(y-4)^2-0.01)((x-179/41)^2+(y-94/41)^2-0.01)=0 [-1.985, 9.116, -0.99, 4.56]}

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