Conic Sections Basics
A free College Algebra lesson from the “Relating Algebra and Geometry” unit, with a worked example and practice problems including step-by-step solutions.
Conic sections are curves created by slicing cones. In College Algebra, the main goal is recognizing equation patterns and interpreting basic features.
What you'll learn
- Recognize circles, parabolas, ellipses, and hyperbolas
- Connect equations to shapes
- Interpret basic conic features
Worked example
Problem. Which conic is represented by (x - 1)^2 + (y + 2)^2 = 16?
- Both x and y are squared.
- The squared terms have the same coefficient.
- This is a circle with radius 4.
Answer: circle
Practice problems
1. An equation with only x squared, such as y = x^2 + 3, is a...
Choices: Parabola · Circle · Ellipse · Line
Show solution
- Warm-up: First identify exactly what the question is asking: An equation with only x squared, such as y = x^2 + 3, is a...
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- One squared variable points to a parabola.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Parabola
2. x^2 + y^2 = 25 represents a...
Choices: Circle · Line · Hyperbola · Cubic
Show solution
- Core Practice: First identify exactly what the question is asking: x^2 + y^2 = 25 represents a...
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Equal squared terms.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Circle
3. x^2 - y^2 = 9 is a basic pattern for a...
Choices: Hyperbola · Circle · Line · Absolute value graph
Show solution
- Challenge: First identify exactly what the question is asking: x^2 - y^2 = 9 is a basic pattern for a...
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- A difference of squared terms suggests a hyperbola.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Hyperbola
4. For x^2 + y^2 = 36, what is the radius?
Show solution
- Circles: First identify exactly what the question is asking: For x^2 + y^2 = 36, what is the radius?
- For circle problems, connect the formula or theorem to the given radius, diameter, chord, arc, or center information.
- The radius squared is 36.
- Check the result by substituting or estimating: the response should match 6 and make sense in the original problem.
Answer: 6
5. The graph of y = x^2 opens...
Choices: Up · Down · Left · Right
Show solution
- Parabolas: First identify exactly what the question is asking: The graph of y = x^2 opens...
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- The leading coefficient is positive.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Up
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