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Conic Sections cheat sheet - grade 10-12

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Math Grade 10-12

Conic Sections Cheat Sheet

A printable reference covering circles, parabolas, ellipses, hyperbolas, standard equations, foci, vertices, asymptotes, and eccentricity for grades 10-12.

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Conic sections are curves formed by slicing a double cone with a plane, and they include circles, parabolas, ellipses, and hyperbolas. Students need this cheat sheet to compare the equations, graphs, and key features of each conic quickly. It is especially useful when converting between standard form and graph features such as center, vertex, focus, and asymptotes.

The most important idea is that each conic has a standard equation that reveals its shape and location. Circles use equal squared terms, parabolas use one squared variable, ellipses use a sum of squared terms, and hyperbolas use a difference of squared terms. Key values such as aa, bb, cc, hh, and kk determine vertices, foci, radius, direction, and asymptotes.

Key Facts

  • A circle with center (h,k)(h,k) and radius rr has equation (x−h)2+(y−k)2=r2(x-h)^2+(y-k)^2=r^2.
  • A vertical parabola has equation (x−h)2=4p(y−k)(x-h)^2=4p(y-k), vertex (h,k)(h,k), focus (h,k+p)(h,k+p), and directrix y=k−py=k-p.
  • A horizontal parabola has equation (y−k)2=4p(x−h)(y-k)^2=4p(x-h), vertex (h,k)(h,k), focus (h+p,k)(h+p,k), and directrix x=h−px=h-p.
  • An ellipse centered at (h,k)(h,k) with horizontal major axis has equation (x−h)2a2+(y−k)2b2=1\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1, where a>ba>b and c2=a2−b2c^2=a^2-b^2.
  • A hyperbola centered at (h,k)(h,k) opening left and right has equation (x−h)2a2−(y−k)2b2=1\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1, vertices (h±a,k)(h\pm a,k), and c2=a2+b2c^2=a^2+b^2.
  • A hyperbola centered at (h,k)(h,k) opening up and down has equation (y−k)2a2−(x−h)2b2=1\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1, vertices (h,k±a)(h,k\pm a), and c2=a2+b2c^2=a^2+b^2.
  • For an ellipse or hyperbola, eccentricity is e=cae=\frac{c}{a}, with 0<e<10<e<1 for ellipses and e>1e>1 for hyperbolas.
  • The asymptotes of (x−h)2a2−(y−k)2b2=1\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1 are y−k=±ba(x−h)y-k=\pm \frac{b}{a}(x-h).

Vocabulary

Conic section
A conic section is a curve formed by intersecting a plane with a double cone.
Focus
A focus is a fixed point used to define a conic, often paired with distance rules involving the curve.
Directrix
A directrix is a fixed line used with a focus to define a parabola by equal distances.
Vertex
A vertex is a turning point or endpoint of a main axis of a conic.
Major axis
The major axis of an ellipse is its longest central axis, with length 2a2a.
Asymptote
An asymptote is a line that a hyperbola approaches but never reaches as the graph extends.

Common Mistakes to Avoid

  • Confusing ellipse and hyperbola signs is wrong because ellipses use a sum, such as (x−h)2a2+(y−k)2b2=1\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1, while hyperbolas use a difference.
  • Using the larger denominator as a2a^2 for every conic is wrong because for hyperbolas, a2a^2 is always under the positive term, not always the larger denominator.
  • Forgetting to square the radius in a circle is wrong because (x−h)2+(y−k)2=r2(x-h)^2+(y-k)^2=r^2, so a right side of 2525 means r=5r=5, not r=25r=25.
  • Mixing up c2=a2−b2c^2=a^2-b^2 and c2=a2+b2c^2=a^2+b^2 is wrong because ellipses use c2=a2−b2c^2=a^2-b^2, while hyperbolas use c2=a2+b2c^2=a^2+b^2.
  • Ignoring the signs of hh and kk is wrong because (x−h)(x-h) and (y−k)(y-k) show shifts, so (x+3)2(x+3)^2 means h=−3h=-3.

Practice Questions

  1. 1 Find the center and radius of the circle (x−2)2+(y+5)2=49(x-2)^2+(y+5)^2=49.
  2. 2 For the parabola (x+1)2=12(y−3)(x+1)^2=12(y-3), find the vertex, focus, and directrix.
  3. 3 For the hyperbola (x−4)29−(y+2)216=1\frac{(x-4)^2}{9}-\frac{(y+2)^2}{16}=1, find the center, vertices, foci, and asymptotes.
  4. 4 Explain how you can tell from an equation whether a conic is a circle, parabola, ellipse, or hyperbola without graphing it.

Understanding Conic Sections

A useful way to understand these curves is through distance. A circle contains every point the same distance from one fixed point. A parabola contains points equally distant from a focus and a straight line called a directrix.

An ellipse has two foci, and the sum of the distances from any point on the curve to those foci stays constant. A hyperbola has two foci too, but it keeps the absolute difference between those two distances constant.

These definitions explain the shapes better than memorizing equations. They show why a parabola has one continuous branch, an ellipse closes into a loop, and a hyperbola separates into two branches.

When reading an equation, first locate the shifted squared expressions. The number paired with x tells how far the graph moves left or right. The number paired with y tells how far it moves up or down.

Then inspect the signs and denominators. For ellipses, the larger denominator marks the direction of the longer axis. For hyperbolas, the positive squared term marks the direction in which the branches open.

This is a frequent source of errors. Students sometimes choose the larger denominator for the opening direction, but that rule belongs to ellipses, not hyperbolas. For a parabola, the variable that is squared helps identify its axis of symmetry.

If x is squared, the graph moves vertically. If y is squared, it moves horizontally.

The values called a, b, and c describe different distances, so they should not be treated as interchangeable labels. In an ellipse, a measures from the center to a vertex on the major axis, while c measures from the center to a focus. The foci stay inside the ellipse.

In a hyperbola, c is larger than a, so the foci lie beyond the vertices. Eccentricity describes how far a conic departs from being circle-like. An ellipse with eccentricity near zero looks nearly circular.

An ellipse with eccentricity near one looks more stretched. A hyperbola has eccentricity greater than one. This distance ratio is useful in astronomy, where planetary paths can be elliptical, and in space science, where some objects follow hyperbolic paths.

Conics appear in technology because their geometric properties control paths of light, sound, and motion. A parabolic reflector sends rays from its focus outward in parallel directions. This is why parabolic shapes are used in satellite dishes, headlights, and some microphones.

Elliptical rooms can carry sound between focal points, which can make whispers travel in surprising ways. Hyperbolas help describe location methods based on differences in signal arrival times. When solving graphing problems, sketch the center or vertex first, then mark vertices and foci before drawing the curve.

For hyperbolas, draw the asymptotes lightly as guides. They show the direction the branches approach, but the branches never meet them.