Find the equation of line parallel to #x-4y=0# that divides the quadrilateral formed by #xy(x-2)(y-3)=0# in two equal areas?

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Answer 1 · 2018-01-07T14:24:44

Equation of the line parallel to #x-4y=0# that divides the quadrilateral in two equal areas is #x-4y+5=0#.

The equation #xy(x-2)(y-3)=0# represents four lines #x=0#, #y=0#, #x=2# and #y=3# forming a rectangle.

It is apparent that vertices of this rectangle are #(0,0),(0,3),(2,0)# and #(2,3)#

and hence center of this rectangle is center of either of the diagonals i.e. #(1,1.5)#

Now any line passing through this point would divided the rectangle in two equal areas.

As the equation of any line parallel to #x-4y=0# is #x-4y=k# and as it passes through #(1,1.5)#, we have #k=1-4xx1.5=1-6=-5#

hence, equation of the line parallel to #x-4y=0# that divides the quadrilateral in two equal areas is #x-4y=-5# or #x-4y+5=0#.

graph{xy(x-2)(y-3)(x-4y)(x-4y+5)((x-1)^2+(y-1.5)^2-0.01)=0 [-3.36, 5.53, -0.616, 3.828]}