Book cover for College Algebra

College Algebra

Michael Sullivan

ISBN #9780321979476

10th Edition

5,609 Questions

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320,854 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces conic sections by starting with the geometry of a right circular cone and explaining how the intersection of a plane with the cone produces various curves such as circles, ellipses, parabolas, and hyperbolas. A detailed focus on parabolas is presented, including their geometric definition involving the focus, directrix, and vertex, and a derivation of the standard equation y² = 4ax. The material emphasizes both foundational geometric concepts and practical applications, ensuring a thorough understanding of conic sections.

Learning Objectives

1

Identify and describe the names and properties of the conic sections, including circles, ellipses, parabolas, and hyperbolas.

2

Understand the construction of a right circular cone and how its intersection with a plane produces various conics.

3

Derive and analyze the equation of a parabola with the vertex at the origin using the distance formula and geometric definitions.

4

Apply geometric concepts such as focus, directrix, vertex, and axis of symmetry in the analysis of conic sections.

Key Concepts

CONCEPT

DEFINITION

Right Circular Cone

A geometric figure generated by rotating a line (generator) about a fixed line (axis) while keeping a constant angle. The point of intersection is the vertex and the surface consists of two parts called nappes.

Conic Sections

Curves produced by the intersection of a right circular cone and a plane. Depending on the angle and position of the intersection, these curves are classified as circles, ellipses, parabolas, or hyperbolas. If the plane passes through the vertex, the results are degenerate conics (a point, a line, or intersecting lines).

Parabola

A set of points in the plane that are equidistant from a fixed point (focus) and a fixed line (directrix). Its axis of symmetry passes through the vertex, which is midway between the focus and the directrix.

Focus

A fixed point used to define a parabola where every point on the parabola is equidistant from the focus and the directrix.

Directrix

A fixed line used with the focus to define a parabola; every point on a parabola is equidistant from the focus and the directrix.

Vertex

The point of intersection between the parabola and its axis of symmetry; it lies midway between the focus and the directrix.

Axis of Symmetry

A line through the focus perpendicular to the directrix; it divides the parabola into two mirror-image halves.

Example Problems

Example 1

The formula for the distance $d$ from $P_{1}=\left(x_{1}, y_{1}\right)$ to $P_{2}=\left(x_{2}, y_{2}\right)$ is $d=$ _______.

Example 2

To complete the square of $x^{2}-4 x,$ add ______.

Example 3

Use the Square Root Method to find the real solutions of $(x+4)^{2}=9 .

Example 4

The point that is symmetric with respect to the $x$ -axis to the point $(-2,5)$ is _______.

Example 5

To graph $y=(x-3)^{2}+1,$ shift the graph of $y=x^{2}$ to the right ______ units and then _______ 1 unit.
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Step-by-Step Explanations

QUESTION

How do you derive the standard equation of a parabola with vertex at (0,0) and focus at (a,0)?

STEP-BY-STEP ANSWER:

Step 1: Recognize that any point P = (x, y) on the parabola is equidistant from the focus F = (a, 0) and the directrix D given by x = -a.
Step 2: Apply the distance formula to set up the equation for the distance from P to F: distance = √((x - a)² + y²).
Step 3: Determine the distance from P to the directrix. Since the directrix is vertical (x = -a), this distance is |x + a|.
Step 4: Set the two distances equal, according to the definition: √((x - a)² + y²) = |x + a|.
Step 5: Square both sides to remove the square root and absolute value: (x - a)² + y² = (x + a)².
Step 6: Expand both sides: x² - 2ax + a² + y² = x² + 2ax + a².
Step 7: Cancel common terms (x² and a²) and simplify: y² = 4ax.
Final Answer: The equation of the parabola is y² = 4ax.

Deriving the Equation of a Parabola with Vertex at the Origin

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Common Mistakes

  • Confusing the generator lines with the axis of the cone.
  • Misidentifying the conic sections when the plane contains the vertex (degenerate conics).
  • Incorrectly applying the distance formula by not considering the absolute value for the distance to the directrix.
  • Forgetting to square both sides when deriving the equation of a parabola, which can lead to errors in sign and coefficients.