Textbook Question
In Exercises 1–18, graph each ellipse and locate the foci. 7x² = 35-5y²
8. Conic Sections
Ellipses: Standard Form
3:45 minutesProblem 23
Textbook Question
In Exercises 19–24, find the standard form of the equation of each ellipse and give the location of its foci. 
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Identify the center of the ellipse, which is at the point (2, 3).
Determine the lengths of the semi-major and semi-minor axes. The semi-major axis extends from (2, 3) to (15, 3) and (-11, 3), giving a length of 13 units. The semi-minor axis extends from (2, 3) to (2, 8) and (2, -2), giving a length of 5 units.
Write the standard form of the ellipse equation: \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), where (h, k) is the center, a is the semi-major axis length, and b is the semi-minor axis length.
Substitute the values into the equation: \( \frac{(x-2)^2}{13^2} + \frac{(y-3)^2}{5^2} = 1 \).
To find the foci, use the formula \( c = \sqrt{a^2 - b^2} \). Calculate c and then determine the coordinates of the foci, which will be at (2+c, 3) and (2-c, 3).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Standard Form of an Ellipse
The standard form of an ellipse's equation is given by (x-h)²/a² + (y-k)²/b² = 1, where (h, k) is the center, 'a' is the semi-major axis, and 'b' is the semi-minor axis. This form helps identify the orientation of the ellipse and its dimensions, which are crucial for graphing and understanding its properties.
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Foci of an Ellipse
The foci of an ellipse are two fixed points located along the major axis, equidistant from the center. The distance from the center to each focus is denoted as 'c', where c² = a² - b². The foci play a significant role in defining the ellipse's shape and are essential for applications in physics and engineering.
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Graphing Ellipses
Graphing an ellipse involves plotting its center, determining the lengths of the semi-major and semi-minor axes, and marking the foci. Understanding the coordinate plane and how to interpret the graph is vital for visualizing the ellipse's properties and solving related problems effectively.
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