Book cover for Algebra and Trigonometry Real Mathematics, Real People

Algebra and Trigonometry Real Mathematics, Real People

Ron Larson

ISBN #9781305071735

7th Edition

6,909 Questions

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234,597 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section covers the equations and properties of circles and parabolas. It emphasizes writing the standard forms of circles and parabolas, computing and identifying key attributes like the center, radius, vertex, focus, and directrix, and applying the reflective property of parabolas in real-world scenarios. Understanding these concepts is essential for solving geometric problems and modeling real-life situations such as highway ramp design and reflective devices.

Learning Objectives

1

Recognize conic sections as intersections of a plane and a double-napped cone.

2

Write and convert equations of circles into standard form.

3

Derive and write equations of parabolas in standard form from given characteristics.

4

Utilize the reflective property of parabolas to solve real-life problems.

5

Sketch graphs of circles and parabolas, identifying centers, radii, vertices, foci, and directrices.

Key Concepts

CONCEPT

DEFINITION

Circle

The set of all points (x, y) in a plane that are equidistant from a fixed point (h, k). Its standard equation is (x − h)² + (y − k)² = r², where r is the radius.

Parabola

The set of all points in a plane that are equidistant from a fixed point called the focus and a fixed line called the directrix. The vertex is the midpoint between the focus and the directrix.

Standard Equation of a Circle

(x − h)² + (y − k)² = r²; when the center is the origin, it simplifies to x² + y² = r².

Standard Equation of a Parabola

For a vertical axis: (x − h)² = 4p(y − k) and for a horizontal axis: (y − k)² = 4p(x − h), where (h, k) is the vertex and p is the directed distance from the vertex to the focus.

Reflective Property of Parabolas

A property stating that a ray coming parallel to the axis of symmetry reflects off the parabola and passes through the focus. This is used in applications such as satellite dishes, flashlights, and telescopes.

Example Problems

Example 1

fill in the blank(s). A _______ is the intersection of a plane and a double-napped cone.

Example 2

fill in the blank(s). A collection of points satisfying a geometric property can also be referred to as a _______ of points.

Example 3

fill in the blank(s). A _______ is the set of all points $$(x, y)$$ in a plane that are equidistant from a fixed point, called the _______ .

Example 4

fill in the blank(s). A _______ is the set of all points $$(x, y)$$ in a plane that are equidistant from a fixed line, called the _______ , and a fixed point, called the _______ , not on the line.

Example 5

What does the equation $(x-h)^{2}+(y-k)^{2}=r^{2}$ represent? What do $h, k,$ and $r$ represent?
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Step-by-Step Explanations

QUESTION

Given a circle with center (-2, -3) and a point on the circle (1, 4), find its standard equation.

STEP-BY-STEP ANSWER:

Step 1: Identify the center (h, k) = (-2, -3).
Step 2: Use the distance formula to compute the radius: r = √[(1 − (−2))² + (4 − (−3))²] = √[(3)² + (7)²] = √(9 + 49) = √58.
Step 3: Substitute h, k, and r into the standard form: (x − h)² + (y − k)² = r² becomes (x + 2)² + (y + 3)² = 58.
Final Answer: (x + 2)² + (y + 3)² = 58.

Standard Circle Equation

QUESTION

Determine the standard equation of a parabola with vertex at the origin and focus at (0, 4).

STEP-BY-STEP ANSWER:

Step 1: Identify the vertex (h, k) = (0, 0) and the focus (0, 4).
Step 2: Calculate p, the distance from the vertex to the focus. Here, p = 4.
Step 3: Since the parabola has a vertical axis, use the form: x² = 4p y.
Step 4: Substitute p = 4 to obtain x² = 16y.
Final Answer: x² = 16y.

Standard Parabola Equation (Vertical Axis)

QUESTION

Find the equation of the tangent line to the parabola y = x² at the point (1, 1).

STEP-BY-STEP ANSWER:

Step 1: Compute the derivative of y = x² to get the slope of the tangent: dy/dx = 2x.
Step 2: Evaluate the derivative at x = 1: slope m = 2(1) = 2.
Step 3: Use the point-slope form of the line with point (1, 1): y − 1 = 2(x − 1).
Step 4: Simplify to get the equation: y = 2x − 1.
Final Answer: y = 2x − 1.

Tangent Line to a Parabola

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Common Mistakes

  • Misidentifying the signs for h and k when converting expressions (e.g., reading (x + 2)² as h = 2 instead of h = -2).
  • Forgetting to square the radius when substituting into the circle's standard equation.
  • Mixing up the formulas for parabolas with vertical and horizontal axes.
  • Overlooking the directed nature of p in parabolic equations, leading to the wrong orientation (upward/downward or left/right).
  • Ignoring the reflective property which requires precise understanding of tangent lines and angles with respect to the focus and the axis.