Conic sections

What the problems look like

A few statements from this skill. Answers, hints and worked solutions live in the practice session.

1. Warm-up

   A conic section is given by $2x^{2} + 6xy + 3y^{2} + 2x + 3y - 8 = 0$. Its type is determined by the discriminant $B^{2} - 4AC$, where $A$, $B$, and $C$ are the coefficients of $x^{2}$, $xy$, and $y^{2}$: negative gives an ellipse, zero a parabola, and positive a hyperbola. Compute the discriminant of this conic.

2. Warm-up

   A conic section is given by $5x^{2} - 4xy + 4y^{2} + x + 4y - 5 = 0$. Its type is determined by the discriminant $B^{2} - 4AC$, where $A$, $B$, and $C$ are the coefficients of $x^{2}$, $xy$, and $y^{2}$: negative gives an ellipse, zero a parabola, and positive a hyperbola. Compute the discriminant of this conic.

3. Warm-up

   A conic section is given by $5x^{2} - 2xy - 2y^{2} + 6x + 4y - 9 = 0$. Its type is determined by the discriminant $B^{2} - 4AC$, where $A$, $B$, and $C$ are the coefficients of $x^{2}$, $xy$, and $y^{2}$: negative gives an ellipse, zero a parabola, and positive a hyperbola. Compute the discriminant of this conic.

Problems shown under their original terms: Generated (template, solver-computed).