Conic Sections
Introduced| English | Meaning |
|---|---|
| ellipse/ɪˈlɪps/ | An oval curve; the set of points whose distances to two foci add to a constant. |
| parabola/pəˈræbələ/ | The curve formed by points equally distant from a fixed point and a fixed straight line. |
| hyperbola/haɪˈpɜːbələ/ | A curve whose points have a constant absolute difference between their distances from two fixed points. |
| conic section/ˈkɒnɪk ˈsekʃn/ | A curve where a plane cuts a cone. |
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…
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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).
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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…
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Every point on an ellipse has the same total distance to the two foci — that constant defines the shape.
Select all the conic sections.
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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.
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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 $x$.
Write from a distance of 4
- Every point on a circle is the same distance from the centre.
- The radius is 4.
A circle is defined as the set of points at a fixed distance from a centre. If that distance is 4, how far is a point on the circle from the centre?
The fixed distance is the radius.
How far is a point on this circle from the centre?
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The defining distance is 4.