Weighted unit circle

Prove that the unit circle, for an inner product on lR^2 is defined as the set of all vectors of unit length ||v|| = 1, of the non-standard inner product [tex]v_1 w_1-v_1 w_2 - v_2 w_1 + 4 v_2 w_2[/tex] is an ellipse.

I know that norm squared will be [tex](v_1 w_1-v_1 w_2 - v_2 w_1 + 4 v_2 w_2) (v_1 w_1-v_1 w_2 - v_2 w_1 + 4 v_2 w_2)[/tex], but I don't really want to multiply that all out to show that it looks like an ellipse. Is there a better way, maybe manipulating the inner product somehow?

 

I figured out another way. Take the dot product (l_2) in |R^2 and compare it with the l_infty inner product.

[tex] B_2 = set(v \in lR^2 | v_1^2 + v_2^2 = 1)[/tex]
[tex] B_\infty = set(v \in lR^2 | max(|v_1|,|v_2|) = 1)[/tex]

Everything in between will be an ellipse.