Formula sheet: Conic Sections
Conic Sections18 min readLesson 1 of 7
1.1 Conics from a cone
A conic section is the curve obtained when a plane cuts a double nappe cone. A horizontal cut gives a circle, a cut parallel to a generator gives a parabola, a slanted cut that enters both nappes gives a hyperbola, and the remaining slanted cuts give ellipses. The degenerate cases are a point, a straight line and a pair of intersecting lines, obtained when the plane passes through the vertex.
The conics are also defined by focus and directrix. A parabola is the set of points equidistant from a fixed point and a fixed line; an ellipse is the set of points whose distances from two fixed points add to a constant; a hyperbola is the set of points whose distances from two fixed points differ by a constant; a circle is the set of points at a fixed distance from a fixed point.
1.2 Standard equations
The standard circle with centre at the origin has equation x squared plus y squared equal to r squared, and the circle with centre (h, k) is (x minus h) squared plus (y minus k) squared equal to r squared. The standard parabola y squared equal to 4ax opens to the right with vertex at the origin and focus at (a, 0). The standard ellipse x squared by a squared plus y squared by b squared equal to 1 has centre at the origin with semi-major axis a and semi-minor axis b. The standard hyperbola x squared by a squared minus y squared by b squared equal to 1 has two branches and opens left and right.
For the ellipse and the hyperbola, c squared is a squared minus b squared and a squared plus b squared respectively, and the eccentricity is c by a. Eccentricity is less than 1 for an ellipse, equal to 1 for a parabola and greater than 1 for a hyperbola. The latus rectum of the parabola is 4a; for the ellipse and hyperbola it is 2b squared by a.
1.3 Useful measurements
| Curve | Eccentricity | Latus rectum | Foci |
|---|---|---|---|
| Circle | 0 | not defined | centre only |
| Parabola y squared equal to 4ax | 1 | 4a | (a, 0) |
| Ellipse x squared by a squared plus y squared by b squared equal to 1 | c by a, with c squared equal to a squared minus b squared | 2b squared by a | (plus or minus c, 0) |
| Hyperbola x squared by a squared minus y squared by b squared equal to 1 | c by a, with c squared equal to a squared plus b squared | 2b squared by a | (plus or minus c, 0) |
In the general second degree equation Ax squared plus Bxy plus Cy squared plus Dx plus Ey plus F equal to 0, the conic type is decided by B squared minus 4AC: negative for an ellipse or circle, zero for a parabola and positive for a hyperbola. This test is useful when the equation is not in standard form.
Exam
Write the equation in standard form first and read a and b from it. Most marks are lost by using b squared plus a squared instead of a squared minus b squared for the ellipse, or the reverse for the hyperbola.