Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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**Problem Statement:** Consider the equation below. \[ r = \frac{5}{2 + 3\sin(\theta)} \] **Tasks:** (a) **Find the eccentricity.** \[ e = \] (b) **Identify the conic.** - Ellipse - Parabola - Hyperbola - None of the above (c) **Give an equation of the directrix (in Cartesian coordinates).** \[ \text{Equation:} \] (d) **Sketch the conic.** **Graph Analysis:** 1. **Top Left Graph:** - This graph appears to show a parabola opening upwards. The vertex is at the origin (0,0), and the parabola is symmetrical across the y-axis. The directrix is shown as a horizontal dashed line below the vertex. 2. **Top Right Graph:** - This graph illustrates a parabola opening to the right. The vertex is at the origin, and the parabola is symmetrical across the x-axis. The directrix is represented as a vertical dashed line to the left of the vertex. 3. **Bottom Left Graph:** - This is a graph of a hyperbola with branches opening left and right. The asymptotes are shown as dashed diagonal lines intersecting at the origin. 4. **Bottom Right Graph:** - The graph shows a hyperbola with vertical and horizontal branches. The asymptotes are diagonal dashed lines that cross at the origin, indicating the center of the hyperbola. Select the correct graph according to the conic section determined from the given equation.
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Transcribed Image Text:**Problem Statement:** Consider the equation below. \[ r = \frac{5}{2 + 3\sin(\theta)} \] **Tasks:** (a) **Find the eccentricity.** \[ e = \] (b) **Identify the conic.** - Ellipse - Parabola - Hyperbola - None of the above (c) **Give an equation of the directrix (in Cartesian coordinates).** \[ \text{Equation:} \] (d) **Sketch the conic.** **Graph Analysis:** 1. **Top Left Graph:** - This graph appears to show a parabola opening upwards. The vertex is at the origin (0,0), and the parabola is symmetrical across the y-axis. The directrix is shown as a horizontal dashed line below the vertex. 2. **Top Right Graph:** - This graph illustrates a parabola opening to the right. The vertex is at the origin, and the parabola is symmetrical across the x-axis. The directrix is represented as a vertical dashed line to the left of the vertex. 3. **Bottom Left Graph:** - This is a graph of a hyperbola with branches opening left and right. The asymptotes are shown as dashed diagonal lines intersecting at the origin. 4. **Bottom Right Graph:** - The graph shows a hyperbola with vertical and horizontal branches. The asymptotes are diagonal dashed lines that cross at the origin, indicating the center of the hyperbola. Select the correct graph according to the conic section determined from the given equation.
The image contains four graphs arranged in a 2x2 grid, each representing different conic sections with their respective axes labeled. **Top Left Graph:** - This is a parabola opening upwards. - Its vertex is at the origin (0,0) on the coordinate plane. - The x-axis and y-axis are marked with values ranging approximately from -10 to 10. **Top Right Graph:** - This graph displays a hyperbola opening horizontally. - The center of the hyperbola is at the origin. - The x-coordinate values range from approximately -10 to 10, and the y-coordinate values range from -5 to 5. **Bottom Left Graph:** - This is a hyperbola with its transverse axis in the vertical direction. - The graph shows two branches opening upwards and downwards. - The axes are labeled with x ranging from approximately -10 to 10 and y from -10 to 10. **Bottom Right Graph:** - Another hyperbola graph, with branches opening to the right and left. - The center is near the origin with asymptotes approaching diagonal lines. - The axes range between x: -10 to 10, y: -10 to 10. Below these graphs, there are selectable radio buttons which likely serve as multiple-choice options for understanding or identifying the conic sections. At the bottom of the image, a "Submit Answer" button is visible, indicating an interactive element likely for assessment purposes.
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Transcribed Image Text:The image contains four graphs arranged in a 2x2 grid, each representing different conic sections with their respective axes labeled. **Top Left Graph:** - This is a parabola opening upwards. - Its vertex is at the origin (0,0) on the coordinate plane. - The x-axis and y-axis are marked with values ranging approximately from -10 to 10. **Top Right Graph:** - This graph displays a hyperbola opening horizontally. - The center of the hyperbola is at the origin. - The x-coordinate values range from approximately -10 to 10, and the y-coordinate values range from -5 to 5. **Bottom Left Graph:** - This is a hyperbola with its transverse axis in the vertical direction. - The graph shows two branches opening upwards and downwards. - The axes are labeled with x ranging from approximately -10 to 10 and y from -10 to 10. **Bottom Right Graph:** - Another hyperbola graph, with branches opening to the right and left. - The center is near the origin with asymptotes approaching diagonal lines. - The axes range between x: -10 to 10, y: -10 to 10. Below these graphs, there are selectable radio buttons which likely serve as multiple-choice options for understanding or identifying the conic sections. At the bottom of the image, a "Submit Answer" button is visible, indicating an interactive element likely for assessment purposes.
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