Show that the equation of the perpendicular bisector of the points (t , t+1) and (3t , t+3) is y+tx=2t^2+t+2. If this perpendicular bisector passes through the points (5 , 2), what is the unknown value of t?

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Answer 1 · 2018-04-03T15:15:34

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The mid point of the two points:

#(3t+t)/2# and #((t+1)+(t+3))/2#

#(4t)/2# and #(2t+4)/2#

#2t# and #t+2# #=>#. #(2t,t+2)#

The gradient/slope of the line between the two points:

#((t+3)-(t+1))/(3t-t)# =#(t+3-t-1)/(2t)#=#2/(2t)#=#1/t#

The gradient of the line that is perpendicular to this will have a gradient of #-t#

The line is of the form #y=-t+c# and it goes through the midpoint #(2t,t+2)#
Substitute these values in:

#t+2=-t##xx##2t+c#

#t+2=-2t^2+c#

#2t^2+t+2=c#

So the line is #y=-tx+2t^2+t+2#

#y+tx=2t^2+t+2#

If the line passes through (5,2)
#=># #2+5t=2t^2+t+2#

#=># #2t^2-4t=0#

#=># #2t(t-2)=0#

#=># #t=0 or t=2#