Master the degree-2 curves: circles, ellipses, parabolas, and hyperbolas. See how they all arise from slicing a cone.
Circles, ellipses, parabolas, and hyperbolas may look different, but they share a beautiful secret: they're all slices of a cone. This is why they're called conic sections.
The discriminant B² - 4AC tells you which conic you have. This single number classifies every degree-2 curve: negative for ellipses, zero for parabolas, and positive for hyperbolas.
Adjust the coefficients of the general conic equation Ax² + Bxy + Cy² + Dx + Ey + F = 0. Watch how the discriminant B² - 4AC determines the curve type.
A = C, B = 0. All points equidistant from center.
See why they're called conic sections. A plane cutting through a double cone creates different curves depending on its angle.
Drag to rotate the view. The colored plane slices through a double cone, creating different conic sections depending on the angle of the cut.
Each conic has a focus-directrix definition. Move your cursor over the curves to see their defining geometric properties in action.
Move your mouse over the curve to see the defining property. Purple dots are foci (or focus for parabola). The dashed line is the directrix.
You now understand the complete classification of conic sections:
Next: We'll discover Bezout's Theorem — the beautiful rule that tells us exactly how many points two algebraic curves share.