finding $\lambda$ when equation of parabola is given

Question

If the equation $\lambda x^2 + 4xy + y^2 + \lambda x + 3y + 2 = 0$ represents a parabola.

Then find $\lambda$.

I got stuck in this question while solving parabola. Is here anybody who can help me ???

Answer 1

Your conic is a parabola if $B^2 - 4AC = 0$ with $B = 4$, $C = 1$, and $A = \lambda$. Thus: $4^2 - 4\lambda = 0$, giving $\lambda = 4$