Precalculus uses eccentricity, e = c/a, both to describe a conic and to build one: find it for an ellipse from its equation, or use a given e to recover a missing a, b or c. For an ellipse c² = a² - b², and for a hyperbola c² = a² + b².
Also taught in Algebra 2, at its own level and with its own examples.
Select your difficulty
Pick more than one for a sheet that mixes them.
The preview can look cramped on a phone. The PDF and printed sheet come out normal.
A hyperbola 144x2−ky2=1 has eccentricity 45.Find k
a² = 144, so a = 12.
e = c/a, so c = 12 × 5/4 = 15.
For a hyperbola c² = a² + b², so k = 225 - 144.
The answer is 81.
Where students go wrong
Dividing by a² instead of a. In x²/400 + y²/256 = 1, a is 20 and c is √(400 - 256) = 12, so e = 12/20 = 3/5; using 400 or 256 in place of the lengths gives the wrong e.
Also called eccentricity of an ellipse or eccentricity of a hyperbola.
What you can put on this worksheet
Eccentricity of an ellipse or hyperbola
What is the eccentricity e=ac of the hyperbola 576y2−100x2=1?→ 1213
Circles and parabolas, translated and general forms, write from e
Find the eccentricity: 49(x+5)2−(y+8)2=1→ e=752
Every property from general form
Give the center, vertices, foci, asymptotes, direction, axis lengths, latus rectum and eccentricity: x2−25y2+50y−50=0→ Center (0,1)Vertices (5,1),(−5,1) Foci (26,1),(−26,1) Asymptotes y=51x+1,y=−51x+1 Opens left and right (horizontal) Transverse axis 10Conjugate axis 2Latus rectum 52e=526
Questions about these worksheets
Yes. Every eccentricity sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.
Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.
Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.
Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.
Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.
That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the eccentricity questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.
Yes. There are two levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.
You pick from 3: eccentricity of an ellipse or hyperbola, circles and parabolas, translated and general forms, write from e and every property from general form. Tick as many as you want and set how many of each, or let it spread them evenly.
Using the wrong relationship. An ellipse has a squared equals b squared plus c squared, and a hyperbola has c squared equals a squared plus b squared. Swapping them gives an ellipse an eccentricity above 1, which is impossible and worth noticing.