what does this graph represent

Question

The graph of the equation $x^2 - xy - 2y^2 = 0$ is...

B) 2 intersecting lines.

I chose this answer following my 'gut feelings' on a MCQ, and it turned out to be right, but when I try to prove myself it's actually quite difficult and I get mixed up... Can someone give me a hint?

Answer 1

Hint:

$$x^2-xy-2y^2=x(x-2y)+y(x-2y)=?$$

Answer 2

You are correct, it can be rewritten in the following form $$(x+y)(x-2y)=0$$ so you have a pair of lines $y=-x$ and $2y=x$

Answer 3

It’s clearly a degenerate conic of some sort. There are no linear terms, so it’s centered at the origin, and it also includes the origin as a solution. The discriminant is $B^2-4AC=(-1)^2-4(1)(-2)=9\gt0$, so it’s a degenerate hyperbola, i.e., a pair of intersecting lines.