Statement $1$: If the area of the triangle bounded by the lines $y=x, x+y=8$ and the line through $P=(h,k)$ parallel to the $x$-axis is $4h^2,$ then $P$ lies on either of the two lines represented by $4x^2+8y-y^2-16=0.$

Statement $2$: The area of the triangle bounded by the lines $y=x, x+y=8$ and the $x$-axis is equal to half of the area of the triangle bounded by the line $x+y=8$ and the two coordinate axes.

Which of the following is true of the above two statements?