Conic Sections: Circles, Parabolas, Ellipses, Hyperbolas
A free Algebra II lesson from the “Equations, Rational Functions, and Conics” unit, with a worked example and practice problems including step-by-step solutions.
Conic sections are the four curves you get by slicing a double cone with a plane. Standard forms: Circle (x - h)^2 + (y - k)^2 = r^2; Parabola y = a(x - h)^2 + k; Ellipse (x - h)^2 / a^2 + (y - k)^2 / b^2 = 1; Hyperbola (x - h)^2 / a^2 - (y - k)^2 / b^2 = 1. The plus-vs-minus between the squared terms is the key signal: plus -> ellipse (or circle if a = b), minus -> hyperbola.
What you'll learn
- Identify a conic from its standard-form equation
- Read center, radius, vertex, or axis lengths from standard forms
- Recognize the four conic sections as slices of a double cone
Worked example
Problem. Identify the conic given by (x - 3)^2 + (y + 1)^2 = 16. State its center and radius.
- Standard form (x - h)^2 + (y - k)^2 = r^2 matches a circle.
- h = 3, k = -1, so center is (3, -1).
- r^2 = 16, so r = 4.
Answer: Circle, center (3, -1), radius 4
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.
Practice problems
1. Identify x^2 + y^2 = 25.
Choices: Circle · Parabola · Ellipse · Hyperbola
Show solution
- Both variables squared with equal coefficients on the same side equal to a positive number.
Answer: Circle
2. Identify y = x^2.
Choices: Circle · Parabola · Ellipse · Hyperbola
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- Only x is squared, y is to the first power.
Answer: Parabola
3. Identify x^2/9 + y^2/16 = 1.
Choices: Circle · Parabola · Ellipse · Hyperbola
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- Both squared, same sign, different denominators.
Answer: Ellipse
4. Identify x^2/9 - y^2/16 = 1.
Choices: Circle · Parabola · Ellipse · Hyperbola
Show solution
- Opposite signs between the squared terms.
Answer: Hyperbola
5. x^2 + y^2 = 49. Radius?
Show solution
- r = sqrt(49) = 7.
Answer: 7
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