Conic Sections
Introduced| English |
|---|
| ellipse/ɪˈlɪps/ |
| parabola/pəˈræbələ/ |
| hyperbola/haɪˈpɜːbələ/ |
| conic section/ˈkɒnɪk ˈsekʃn/ |
Slicing a cone
- Take a double cone and slice it with a flat plane at different tilts.
- Cut straight across and you get a circle; tilt a little and it stretches to an ellipse.
- Tilt more and it opens into a parabola; steeper still and it splits into a hyperbola.
- These four curves — the conic sections — appear all over science and design.
The family of conics
- A conic section 圆锥曲线 is a curve where a plane cuts a cone.
- The four types are the circle, the ellipse, the parabola, and the hyperbola.
- Each has a clean implicit equation and a geometric "distance" definition.
- Planets orbit on ellipses; thrown balls and satellite dishes trace parabolas.
The conic sections are the curves formed by…
A conic section is the intersection of a plane and a cone; the slice angle gives a circle, ellipse, parabola, or hyperbola.
A circle is a special conic section (an ellipse whose two foci coincide).
When the two foci merge into one centre, the ellipse becomes a perfectly round circle.
The ellipse
- An ellipse 椭圆 is the set of points whose distances to two fixed foci add to a constant.
- Its equation is $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, a stretched circle.
- A circle is just the special case where the two foci merge into one centre.
- The longer axis is the major axis; the shorter is the minor axis.
An ellipse is the set of points where the sum of distances to two foci is…
Every point on an ellipse has the same total distance to the two foci — that constant defines the shape.
Select all the conic sections.
A triangle is not a conic. The circle, ellipse, parabola, and hyperbola are the conic sections.
Parabola and hyperbola
- A parabola 抛物线 is the set of points equidistant from a focus and a fixed line (the directrix).
- Its equation looks like $y = ax^2$ — the graph of a quadratic.
- A hyperbola 双曲线 is where the difference of distances to two foci is constant.
- Its equation is $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$, opening into two branches.
Which conic section?
The squared terms tell you the conic. Sort each equation.
A ____ is the conic where each point is equidistant from a focus and a directrix line.
A parabola balances distance to a focus against distance to a fixed line (the directrix).
Recognising a conic
- Read the equation's form: an $x^2 + y^2$ with equal coefficients is a circle.
- Different positive coefficients on $x^2$ and $y^2$ give an ellipse.
- Only one squared term gives a parabola; a minus sign between them gives a hyperbola.
- The sign pattern of the squared terms tells you which conic you have.
Not every conic is a function of $x$. Circles, ellipses, and sideways parabolas fail the vertical-line test. Describe them with implicit equations or parametric forms, not by trying to force a single $y = f(x)$.
Identify the conic $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1$.
- Both terms are squared, positive, and different coefficients.
- That is the equation of an ellipse.
- It stretches to $x = \pm 3$ (major axis) and $y = \pm 2$ (minor axis).
The conic sections — circle, ellipse, parabola, and hyperbola — are the curves formed by slicing a cone. Each has a distance definition (an ellipse sums distances to two foci) and a signature implicit equation, and most are not functions of $x$.