An xy term means the conic has been turned. The discriminant still names it whatever the angle, and cot of twice the angle is A minus C over B, which is what turns it back.
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.
Testing the conic with the old rule. Once there is an xy term, comparing A and C tells you nothing; only B squared minus 4AC does, and it is the sign of it that names the conic.
Also called rotation of axes, xy term, discriminant of a conic or identifying conics.
What you can put on this worksheet
The angle of rotation
Find the angle of rotation that removes the xy-term: x2+3xy=9→ 30∘
Naming a rotated conic
Identify the conic: x2+3xy=9→ Hyperbola
Rewrite in x'y' coordinates
Rotate the axes to remove the xy-term: xy=6→ θ=45∘;x′2−y′2=12
Rotate the axes by a given angle
Rotate the axes by 30∘:write x2−y2=5 interms of uand v.→ 2u2−3uv−2v2=5
sqrt 3 cross terms, the rotated equation
Rotate the axes to remove the xy-term, and write the result in standard form: 13x2+63xy+7y2=64→ θ=30∘;4u2+16v2=1
Questions about these worksheets
Yes. Every rotations of conic sections 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 rotations of conic sections 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 three 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 5: the angle of rotation, naming a rotated conic, rewrite in x'y' coordinates, rotate the axes by a given angle and sqrt 3 cross terms, the rotated equation. Tick as many as you want and set how many of each, or let it spread them evenly.
Testing the conic with the old rule. Once there is an xy term, comparing A and C tells you nothing; only B squared minus 4AC does, and it is the sign of it that names the conic.