Textbook Question
Circles
Sketch the circles in Exercises 53–56. Give polar coordinates for their centers and identify their radii.
r = −2 cos θ
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Ch. 11 - Parametric Equations and Polar Coordinates
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 11, Problem 11.6.10
Parabolas
Exercises 9-16 give equations of parabolas. Find each parabola's focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch.
x² = 6y
Verified step by step guidance1
Identify the form of the given parabola equation. The equation \(x^{2} = 6y\) is in the form \(x^{2} = 4py\), which represents a parabola that opens either upward or downward along the y-axis.
Compare the given equation \(x^{2} = 6y\) with the standard form \(x^{2} = 4py\) to find the value of \(p\). Here, \(4p = 6\), so solve for \(p\) by dividing both sides by 4: \(p = \frac{6}{4} = \frac{3}{2}\).
Determine the focus of the parabola. For the form \(x^{2} = 4py\), the focus is located at \((0, p)\). Using the value of \(p = \frac{3}{2}\), the focus is at \((0, \frac{3}{2})\).
Find the equation of the directrix. The directrix is a horizontal line given by \(y = -p\). Substitute \(p = \frac{3}{2}\) to get the directrix equation \(y = -\frac{3}{2}\).
To sketch the parabola, plot the vertex at the origin \((0,0)\), mark the focus at \((0, \frac{3}{2})\), and draw the directrix line \(y = -\frac{3}{2}\). Then, sketch the parabola opening upward, symmetric about the y-axis, passing through points that satisfy \(x^{2} = 6y\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Standard Form of a Parabola
A parabola can be expressed in a standard form such as x² = 4py or y² = 4px, where p represents the distance from the vertex to the focus (and directrix). Recognizing this form helps identify key features like the vertex, focus, and directrix easily.
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Circles in Standard Form Example 1
Focus and Directrix of a Parabola
The focus is a fixed point inside the parabola where all reflected rays converge, and the directrix is a fixed line outside the parabola. The parabola is the set of points equidistant from the focus and the directrix, which defines its shape and position.
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5:28Horizontal Parabolas
Graphing Parabolas
Sketching a parabola involves plotting the vertex, focus, and directrix, then drawing a smooth curve equidistant from the focus and directrix. Understanding the orientation (vertical or horizontal) and scale based on p helps create an accurate graph.
Recommended video:
7:42Properties of Parabolas
Related Practice
Textbook Question
Ellipses
Exercises 25 and 26 give information about the foci and vertices of ellipses centered at the origin of the xy−plane. In each case, find the ellipse's standard−form equation from the given information.
Foci: ( ±√2, 0) Vertices: (±2,0)
Textbook Question
Parabolas
Exercises 9-16 give equations of parabolas. Find each parabola's focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch.
x = −3y²
Textbook Question
Shifting Conic Sections
You may wish to review Section 1.2 before solving Exercises 39-56.
Exercises 53-56 give equations for hyperbolas and tell how many units up or down and to the right or left each hyperbola is to be shifted. Find an equation for the new hyperbola, and find the new center, foci, vertices, and asymptotes.
x²/4 − y²/5 = 1, right 2, up 2
Textbook Question
Hyperbolas
Exercises 27-34 give equations for hyperbolas. Put each equation in standard form and find the hyperbola's asymptotes. Then sketch the hyperbola. Include the asymptotes and foci in your sketch.
8x² − 2y² = 16
Textbook Question
Polar to Cartesian Equations
Replace the polar equations in Exercises 27–52 with equivalent Cartesian equations. Then describe or identify the graph.
r = 3 cos θ
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