Circles, parabolas, ellipses and hyperbolas as conic sections
Conic sections are the curves obtained when a plane cuts a double cone: a circle, an ellipse, a parabola or a hyperbola. A circle with centre (h, k) and radius r has equation (x − h)² + (y − k)² = r². The other conics can be defined using a fixed point called the focus and a fixed line called the directrix: the ratio of a point's distance from the focus to its distance from the directrix is a constant e, the eccentricity. The curve is a parabola when e = 1, an ellipse when e < 1 and a hyperbola when e > 1.
The standard parabola y² = 4ax has its focus at (a, 0), directrix x = −a and a latus rectum of length 4a. The ellipse x²/a² + y²/b² = 1 with a > b has foci at (±c, 0), where c² = a² − b², and the sum of the distances from any point on it to the two foci is 2a. The hyperbola x²/a² − y²/b² = 1 has c² = a² + b², and the difference of the focal distances is 2a. For both the ellipse and the hyperbola, e = c/a and the latus rectum has length 2b²/a.
Why it matters
Conics appear throughout science and engineering: planets move in elliptical orbits, satellite dishes and car headlights use the reflecting property of the parabola, and some navigation systems locate a position using hyperbolas. Identifying a conic from its equation and reading off its focus, vertices and eccentricity is a standard Class 11 and JEE skill.
Key terms
- Focus
- A fixed point used, together with the directrix, to define a conic.
- Directrix
- A fixed line from which distances are measured when defining a conic.
- Eccentricity
- The constant ratio of distance from the focus to distance from the directrix, which fixes the conic's type.
- Latus rectum
- The chord through a focus drawn perpendicular to the main axis of the conic.