Question

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?

Answer 1

`2015x-2012y=4036`

Explanation:

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`