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28. Center: \( (3,2) ; a=3 c \); foci: \( (1,2),(5,2) \) 29. Vertices: \( (0,2),(4,2) \); endpoints of the minor axis: \( (2,3),(2,1) \) 30. Vertices: \( (5,0),(5,12) \); endpoints of the minor axis: \( (1,6),(9,6) \)

Ask by Barnett Bernard. in the United States
Mar 18,2025

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The equations of the ellipses are as follows: 1. **Problem 28:** \(\frac{(x - 3)^2}{36} + \frac{(y - 2)^2}{32} = 1\) 2. **Problem 29:** \(\frac{(x - 2)^2}{4} + (y - 2)^2 = 1\) 3. **Problem 30:** \(\frac{(x - 5)^2}{16} + \frac{(y - 6)^2}{36} = 1\)

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The Deep Dive

In the first case, with a center at \( (3,2) \) and foci at \( (1,2) \) and \( (5,2) \), we can deduce that this describes an ellipse. The distance between the foci gives us a good hint at the relationship between \( a \), \( b \), and \( c \) values, where \( c = 2 \) (half the distance between the foci), leading to \( a = 3 \) and consequently a minor axis \( b \) derived from the equation \( c^2 = a^2 - b^2 \). When it comes to the given vertices and endpoints of the minor axis, these points help build the geometry of the ellipse. For instance, in the second equation, your vertices are on the major axis, while the minor axis is vertical with coordinates indicating the ellipse height. In contrast, the third set illustrates a vertically oriented ellipse, where the major axis is vertical with the center located at the midpoint of the vertices, providing clarity on its shape and position on the coordinate plane.

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