Ch.12 - Parametric and Polar Curves

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch.12 - Parametric and Polar Curves

Problema 12.4.50

Capitolo 12, Problema 12.4.50

39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin.

Guida verificata passo dopo passo

Identify the type of conic section: Since the graph shows a hyperbola centered at the origin with vertices and foci along the y-axis, it is a vertical hyperbola.

Recall the standard form of the equation for a hyperbola centered at the origin with a vertical transverse axis: \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\).

Determine the value of \(a\): The vertices are at \((0, 6)\) and \((0, -6)\), so the distance from the center to each vertex is \(a = 6\), which means \(a^2 = 36\).

Determine the value of \(c\): The foci are at \((0, 10)\) and \((0, -10)\), so the distance from the center to each focus is \(c = 10\), which means \(c^2 = 100\).

Use the relationship between \(a\), \(b\), and \(c\) for hyperbolas: \(c^2 = a^2 + b^2\). Substitute the known values to solve for \(b^2\): \(100 = 36 + b^2\), then \(b^2 = 100 - 36\).

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Standard Equation of a Hyperbola Centered at the Origin

A hyperbola centered at the origin with a vertical transverse axis has the equation \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \). Here, \(a\) is the distance from the center to each vertex along the y-axis, and \(b\) relates to the distance along the conjugate axis. This form is essential to write the equation based on given vertices and foci.

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Percorso guidato

5:59

Graph Hyperbolas NOT at the Origin

Relationship Between Vertices, Foci, and Parameters \(a\), \(b\), and \(c\)

In a hyperbola, \(a\) is the distance from the center to each vertex, and \(c\) is the distance from the center to each focus. These satisfy the equation \( c^2 = a^2 + b^2 \). Knowing \(a\) and \(c\) allows calculation of \(b\), which is necessary to complete the hyperbola's equation.

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Percorso guidato

5:22

Foci and Vertices of Hyperbolas

Graph Interpretation and Coordinate Identification

Analyzing the graph helps identify key points such as vertices and foci coordinates. For this hyperbola, vertices at (0,6) and (0,-6) give \(a=6\), and foci at (0,10) and (0,-10) give \(c=10\). These values are critical inputs for forming the hyperbola's equation.

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05:32

Intro to Polar Coordinates

Pratica correlata

Domanda del libro di testo

Tangent line at the origin Find the polar equation of the line tangent to the polar curve r=4cosθ at the origin. Explain why the slope of this line is undefined.

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Domanda del libro di testo

What is the polar equation of the horizontal line y = 5?

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15–30. Working with parametric equations Consider the following parametric equations.

a. Eliminate the parameter to obtain an equation in x and y.

b. Describe the curve and indicate the positive orientation.

x = r − 1, y = r³; −4 ≤ r ≤ 4

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33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.

The region inside the curve r = √(cos θ)

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Domanda del libro di testo

31–36. Eliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y.

x=2 sin 8t, y=2 cos 8t

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Domanda del libro di testo

37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.

r = 6 cos θ + 8 sin θ

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