# Show that $P(x,y)=0$ is a hyperbola if $b^2−4ac>0$ .

The following exercise is from Thomas Garrity's Algebraic Geometry: A Problem Solving Approach. I write the polynomial $$P(x, y) = ax^2+bxy+cy^2+dx+ey+h$$ in the form $$P(x, y) = Ax^2 + Bx + C$$ where $$A$$, $$B$$, and $$C$$ are polynomial functions of $$y$$. This $$P(x, y) = Q(x)$$ has the discriminant $$\Delta_x(y) = (b^2 − 4ac)y^2 + (2bd − 4ae)y + (d^2 − 4ah)$$. And I stuck!

Any hint to start solving this?

• Isn’t this more or less the same problem that you’ve already attacked in previous questions with different values of the discriminant? – amd Nov 24 '18 at 22:51
• @amd, Some remarks: 1- I am a member of MSE for less than a month, and my experience to this day is that anytime I've tried to ask only a little more/different question I've been replied like why don't you ask it as a new post? And this question would've been much longer if I had posted all three discriminant cases in one single question so most probably even worse replies.. Also, answers (= resulting sets to be explained) are different. 2- I introduced $\Delta_x(y)$, etc for better clarifications because may not anyone who reads this question reads also any other previous questions. – 72D Nov 24 '18 at 23:06