A line segment is bisected by a line with the equation # - 3 y + 5 x = 2 #. If one end of the line segment is at #( 7 , 9 )#, where is the other end?

1 Answer
Aditya Banerjee.
Nov 4, 2016

Long explanation !!

Explanation:

Slope #m_1# (say) of the line #-3y+5x=2# is :
#-3y=-5x+2# ...............(i)
#:.y=5/3x-2/3#
#:.m_1=5/3.#

Let slope of the line whose one end is #(7,9)# be #m_2# (say).
#:.m_1xxm_2=-1.# [The two lines are perpendicular to each other].

#:.5/3xxm_2=-1#
#:.m_2=-3/5.#

#:.# The equation of the line whose one end is #(7,9)# is :
#(y-y_1)=m(x-x_1)#
#:.# #:.(y-9)=-3/5(x-7)#
#:.3x+5y=66.# is the equation. ..............(ii)

Now, solving equations (i) & (ii), we get the value #(x,y)# which represents the midpoint of the line whose one end is #(7,9)#.

Now to find out the coordinates say, #(a,b)# of the other end, use Distance-Section formula.

#:.d=sqrt[(y_2-y_1)^2+(x_2-x_1)^2#.

#:.sqrt[(x-a)^2+(y-b)^2]=sqrt[(x-7)^2+(y-9)^2.#

Here, #(x,y)# is midpoint & #(a,b)# is the coordinate of the other required end.

Now, I leave it to you. Just put the values and solve.
Best of Luck.

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