The four vertices of a parallelogram are A (2,4), B (8,4), C (0,0), D (6,0). Point P is (2016, 2017). Line L passes through P an divides parallelogram ABCD into two regions of equal area. What is the equation of line L in standard form?

1 Answer
Nov 26, 2017

2015x-2012y=4036

Explanation:

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Assume that the line L that passes through P(2016,2017) cuts AB at E and CD at F, respectively, as shown in the figure.
let AE=b, and EB=a,
given that L divides the parallelogram ABCD into two regions of equal area, => CF=a, and FD=b,
=> a+b=6
=> a=6-b
=> E(x_E,y_E)=(2+b,4)
=> F(x_F, y_F)=(a,0)
=> slope of FE=m_(FE)=4/(2+b-a)=4/(2+b-6+b)=2/(b-2)
=> slope of FP=m_(FP)=(2017)/(2016-a)=(2017)/(2016-6+b)=(2017)/(2010+b)
As m_(FE)=m_(FP),
=> 2/(b-2)=(2017)/(2010+b)
=> b=8054/2015
=> a=6-8054/2015=4036/2015
=> slope =m=2/(b-2)=2/((8054/2015)-2)=2015/2012
Hence, equation of the line passing through (a,0)=(4036/2015,0) with a slope of m=2015/2012 in slope-intercept form is :
y=2015/2012(x-4036/2015)
Hence, equation of L in standard form is :
2015x-2012y=4036