Conic sections are curves formed by slicing a double cone with a plane. The four main conics are circles, ellipses, parabolas, and hyperbolas, and each appears in many areas of math and science. They describe paths, shapes, mirrors, orbits, and design curves.
Understanding conics helps students connect geometry, algebra, and real-world modeling.
Key Facts
- Circle standard form: (x - h)^2 + (y - k)^2 = r^2
- Ellipse standard form: (x - h)^2/a^2 + (y - k)^2/b^2 = 1
- Hyperbola standard form: (x - h)^2/a^2 - (y - k)^2/b^2 = 1 or (y - k)^2/a^2 - (x - h)^2/b^2 = 1
- Parabola standard form: (x - h)^2 = 4p(y - k) or (y - k)^2 = 4p(x - h)
- General second-degree form: Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
- If B = 0, then A = C gives a circle, A and C same sign gives an ellipse, exactly one squared variable gives a parabola, and A and C opposite signs gives a hyperbola.
Vocabulary
- Conic section
- A conic section is a curve formed by the intersection of a plane and a double cone.
- Focus
- A focus is a fixed point used to define a conic by distances from points on the curve.
- Directrix
- A directrix is a fixed line used with a focus to define a parabola and other conics by distance ratios.
- Eccentricity
- Eccentricity is a number that measures how much a conic differs from a circle.
- Asymptote
- An asymptote is a line that a curve approaches more and more closely without becoming the curve.
Common Mistakes to Avoid
- Calling every oval an ellipse without checking the equation. An ellipse has two squared terms with the same sign and unequal coefficients in standard form.
- Forgetting to complete the square before identifying the center. Terms like x^2 - 6x and y^2 + 4y must be rewritten to reveal h and k.
- Mixing up hyperbolas and ellipses because both have two squared variables. A hyperbola has squared terms with opposite signs, while an ellipse has squared terms with the same sign.
- Using p as the vertex instead of the focal distance in a parabola. In (x - h)^2 = 4p(y - k), the vertex is (h, k) and p tells the distance and direction to the focus.
Practice Questions
- 1 Identify the conic and its center or vertex: (x - 3)^2/16 + (y + 2)^2/9 = 1.
- 2 Rewrite x^2 + y^2 - 8x + 6y - 11 = 0 in standard form, then identify the conic and its radius.
- 3 A plane slices one nappe of a cone parallel to a side of the cone. Explain which conic is formed and why its geometric definition matches that slice.