Question

x²+2 x y+y²+x-y=0 Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola.

Answer 1

After rewriting the given equation in the general conic section form and calculating the discriminant, which is zero, we determine that the graph of the equation represents a parabola.

Step-by-step explanation:

To use the discriminant to determine whether the graph of the equation x²+2xy+y²+x-y=0 is a parabola, an ellipse, or a hyperbola, we must first rewrite the equation in the general quadratic form ax²+bxy+cy²+dx+ey+f=0. In this case, we have a=1, b=2, c=1, d=1, e=-1, and f=0. The discriminant Δ for a conic section is given by Δ=b²-4ac.

Substituting the given values, we get Δ=(2)²-4(1)(1)=4-4=0. The discriminant being zero implies that the graph represents a parabola, according to conic section theory.