2016 AMC 12A Problem 7

Question

Which of these describes the graph of

x^2(x+y+1)=y^2(x+y+1)?

two parallel lines

two intersecting lines

three lines that all pass through a common point

three lines that do not all pass through a common point

a line and a parabola

Move everything to one side and factor out

x+y+1

(x-y)(x+y)(x+y+1)=0

is the union of three lines

Moving all terms to one side gives

(x^2-y^2)(x+y+1)=0,

which factors as

(x-y)(x+y)(x+y+1)=0.

The graph is therefore the union of the lines

x=y,

x=-y,

and

x+y+1=0.

The first two lines intersect at the origin, but the third line

x+y=-1

is parallel to

x+y=0

and does not pass through the origin. So the graph consists of three lines that do not all pass through a common point.

Thus, the correct answer is D.

Problem 7 in other years