The distance between the lines represented by x2+2xy+y2-8mx-8my-9m2=0 is?

1 Answer
Apr 20, 2018

The given equation of lines

#x^2+2xy+y^2-8mx-8my-9m^2=0#

#=>(x+y)^2-8m(x+y)-9m2=0#

#=>(x+y)^2-9m(x+y)+m(x+y)-9m2=0#

#=>(x+y)(x+y-9m)+m(x+y-9m)=0#
#=>(x+y-9m)(x+y+m)=0#

Hence the given equation represent a pair of parallel straight lines having equations

#(x+y-9m)=0#

#=>1/sqrt2x+1/sqrt2y=(9m)/sqrt2#

#=>cos45^@x+sin45^@y=(9m)/sqrt2.....[1]#

And

#(x+y+m)=0#

#=>-1/sqrt2x-1/sqrt2y=m/sqrt2#

#=>cos225^@x+sin225^@y=m/sqrt2.....[2]#

The normal forms of equations of two parallel straight lines suggest that the perpendiculars drawn on them from origin are at #180^@#as #(225^@-45^@)=180^@#

So the distance between the lines will be

#PQ=OP+OQ=(9m)/sqrt2+m/sqrt2=(10m)/sqrt2=5sqrt2m#

A graphical representation of the problem for #m=1#
enter image source here