Conic sections are curves formed by slicing a double cone with a plane. This cheat sheet helps students recognize circles, parabolas, ellipses, and hyperbolas from equations, graphs, and key features. Worked examples are useful because conics often look similar until you identify the squared terms, signs, and centers.
A clear reference makes it easier to move between standard form, graph features, and geometric meaning.
The core ideas are standard forms, completing the square, and matching equations to graphs. Circles and ellipses have both and terms with the same sign, while hyperbolas have opposite signs. Parabolas have only one squared variable and are described by a vertex, focus, directrix, and parameter .
For every conic, the goal is to identify the center or vertex first, then use the formula to find the remaining features.
Key Facts
- A circle with center and radius has equation .
- A vertical parabola with vertex has equation , focus , and directrix .
- A horizontal parabola with vertex has equation , focus , and directrix .
- An ellipse centered at with horizontal major axis has equation , where .
- For an ellipse, the foci satisfy , and the foci lie on the major axis.
- A horizontal hyperbola centered at has equation and asymptotes .
- A vertical hyperbola centered at has equation and asymptotes .
- In a general quadratic equation with , same-sign and suggest an ellipse or circle, opposite signs suggest a hyperbola, and exactly one squared term suggests a parabola.
Vocabulary
- Conic section
- A conic section is a curve formed by the intersection of a plane and a double cone.
- Focus
- A focus is a fixed point used to define or locate a conic, such as the point inside a parabola, ellipse, or hyperbola.
- Directrix
- A directrix is a fixed line used with a focus to define a parabola as the set of points equally distant from both.
- Vertex
- A vertex is a key turning point of a conic, such as the endpoint of a parabola or one of the closest points on a hyperbola.
- Major axis
- The major axis is the longer central axis of an ellipse, passing through its center, vertices, and foci.
- Asymptote
- An asymptote is a line that a hyperbola approaches but does not touch as its branches extend.
Common Mistakes to Avoid
- Confusing ellipse and hyperbola signs is wrong because an ellipse has squared terms added, while a hyperbola has one squared term subtracted, such as .
- Forgetting to complete the square is wrong because equations like do not reveal the center until written as .
- Using for an ellipse is wrong because ellipse foci use , while hyperbola foci use .
- Mixing up horizontal and vertical parabolas is wrong because opens up or down, while opens left or right.
- Ignoring the value and sign of is wrong because gives both the distance to the focus and the opening direction of a parabola.
Practice Questions
- 1 Identify the center and radius of the circle .
- 2 For the parabola , find the vertex, value of , focus, and directrix.
- 3 For the hyperbola , find the center, vertices, and asymptotes.
- 4 Explain how you can tell whether an equation represents a circle, ellipse, parabola, or hyperbola by looking at the squared terms and their signs.
Understanding Conic Sections Worked Examples
A worked conic problem usually becomes manageable when you follow the same order every time. First move the constant term away from the terms containing variables. Next group the x terms together and the y terms together.
If a group has a number multiplying its squared term, factor that number out before completing the square. Then complete the square separately for each variable.
The final step is to divide so that the right side becomes one when possible. This order prevents a common mistake where students add a value inside a bracket but forget that the value is affected by a coefficient outside the bracket.
Completing the square is more than an algebra trick. It reveals a shift in the graph. For example, the expression x squared minus six x becomes the quantity x minus three squared after adding nine.
That added nine changes the equation, so it must be balanced elsewhere. When the x and y expressions are both written as squared brackets, the numbers inside the brackets show the horizontal and vertical movement. Students often lose signs here.
A bracket that reads x minus three means a shift right three units. A bracket that reads y plus two means a shift down two units. Reading every bracket slowly is safer than trying to memorize the direction.
The denominators in an ellipse or hyperbola tell you about stretch, not just numbers to copy into a formula. The larger denominator marks the direction with the greater spread. For an ellipse, that direction contains the longest axis and the foci.
For a hyperbola, it tells you whether the branches open left and right or up and down. Hyperbola asymptotes are guides, not parts of the curve. Draw them through the center first, then plot the vertices before sketching the branches.
A correct graph should show the branches approaching the asymptotes without touching them. This visual check can catch a swapped denominator or an incorrect slope.
Conics appear whenever distance rules create shapes. Satellite dishes and car headlights use parabolic reflectors because rays parallel to the axis meet at the focus. Elliptical rooms can concentrate sound between focal points.
Some spacecraft paths are hyperbolas when an object passes a planet quickly enough to escape. In class, focus on connecting algebra to a picture. Mark the center or vertex before finding every other point.
Keep a small sign checklist for opening direction, shifts, and subtraction. For mixed problems, graphing a few key points gives evidence that your classification and calculations fit the equation.