Conic Sections
Easy Overview
If you cut a cone at different angles, you get different shapes — a circle, an ellipse, a parabola, or a hyperbola. Wild, right? These 'conic sections' show up everywhere: the path of a thrown ball (parabola), the orbit of planets (ellipse), the shape of a satellite dish (parabola again). This chapter teaches you how to describe them with equations.
What are conic sections?
Take a cone (like an ice cream cone but double-ended). Slice it straight across and you get a circle. Slice it at a tilt and you get an ellipse. Slice it parallel to the slant and you get a parabola. Slice it steep enough to cut both halves and you get a hyperbola. All four shapes come from the same cone — just different angles. That's why they're called 'conic sections'.
Parabola — the U-shaped curve
A parabola is the set of points equidistant from a fixed point (focus) and a fixed line (directrix). Standard form: y² = 4ax (opens right). The focus is at (a, 0), the directrix is x = −a, and the vertex is at (0, 0). Every parabola is symmetric about its axis. The latus rectum is the chord passing through the focus perpendicular to the axis — its length is 4a.
Ellipse — the stretched circle
An ellipse looks like a squashed circle. Standard form: x²/a² + y²/b² = 1, where a > b. The longer axis is the major axis (length 2a), the shorter is the minor axis (length 2b). There are two foci (plural of focus) — the sum of distances from any point on the ellipse to the two foci is constant (= 2a). Eccentricity e = √(1 − b²/a²) — how 'squashed' it is. e = 0 means a circle.
Hyperbola — the mirrored curve
A hyperbola has two separate branches that mirror each other. Standard form: x²/a² − y²/b² = 1. The minus sign is what makes it different from an ellipse — that changes everything. The eccentricity e = √(1 + b²/a²) is always > 1. Hyperbolas have asymptotes — diagonal lines that the curve approaches but never touches: y = ±(b/a)x.
Key Points
- •All conics come from slicing a double cone
- •Parabola: set of points equidistant from focus and directrix
- •Standard parabola y² = 4ax: focus (a,0), directrix x = −a
- •Ellipse: x²/a² + y²/b² = 1, sum of distances to foci = 2a
- •Eccentricity: e = 0 (circle), 0 < e < 1 (ellipse), e = 1 (parabola), e > 1 (hyperbola)
- •Hyperbola: x²/a² − y²/b² = 1, asymptotes y = ±(b/a)x
- •Latus rectum: chord through focus perpendicular to axis
Practice Questions
- Find the focus, directrix, and latus rectum of y² = 16x.
- Find the equation of an ellipse with foci (±4, 0) and major axis length 10.
- Find the eccentricity and asymptotes of x²/9 − y²/16 = 1.
- A bridge is in the shape of a parabolic arch. The span is 30 m and the maximum height is 10 m. Find the equation of the parabola.
- Prove that for any point on an ellipse, the sum of distances to the foci is constant.