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Chapter 6: Problem 18

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Vertical major axis; passes through the points (0,6) and (3,0)

Short Answer

Expert verified
The standard form of the equation of an ellipse centered at the origin and with a vertical major axis, which also passes through the points (0,6) and (3,0), is \(x^{2} / 9 + y^{2} / 36 = 1\).

Step by step solution

01

Identify the major and minor axes

Firstly, we need to note that an ellipse with a vertical major axis and center at the origin has a standard form equation of \(x^{2} / b^{2} + y^{2} / a^{2} = 1\) where \(2a\) is the length of the major axis and \(2b\) is the length of the minor axis. From the given points, we can gather that the major axis has a length of 12 (as the y-coordinate of the point (0,6) implies that the top point of the major axis is at (0,6) and the bottom point at (0,-6)) and the minor axis has a length of 6 (as the x-coordinate of the point (3,0) implies that the rightmost point of the minor axis is at (3,0) and the leftmost at (-3,0)). So, \(a=6\) and \(b=3\).
02

Substitution of values

Now, substitute the values of \(a\) and \(b\) into the standard form equation. This gives us \(x^{2} / 9 + y^{2} / 36 = 1\).

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