Question

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write the letter of the correct equation on the blank.
4.9) ____ major axis is vertical.
4.10) ____ minor axis is vertical.
a. \\(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\\)
b. \\(\frac{(y - k)^2}{a^2} + \frac{(x - h)^2}{b^2} = 1\\)
(other equations shown: \\(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\\), \\(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\\), \\(e = \frac{\sqrt{a^2 - b^2}}{a}\\), \\(f = \sqrt{a^2 - b^2}\\), \\(\frac{(y - k)^2}{a^2} + \frac{(x - h)^2}{b^2} = 1\\))

Explanation:

Step1: Recall Ellipse Equation Rules

For an ellipse, the standard form depends on the major axis direction. If the major axis is vertical, the \(y\)-term has the larger denominator (\(a^2\) under \(y\)-term). If the minor axis is vertical, the \(x\)-term has the larger denominator (\(a^2\) under \(x\)-term), meaning the major axis is horizontal, and minor is vertical.

Step2: Analyze 4.9 (Major Axis Vertical)

The standard form for an ellipse with vertical major axis is \(\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1\) (since \(a > b\) for major axis length \(2a\)). Looking at options, option B is \(\frac{(y - k)^2}{a^2}+\frac{(x - h)^2}{b^2}=1\), which matches (rewritten order, but \(y\)-term has \(a^2\)).

Step3: Analyze 4.10 (Minor Axis Vertical)

Minor axis vertical means major axis is horizontal, so \(x\)-term has \(a^2\). The standard form is \(\frac{(x - h)^2}{a^2}+\frac{(y - k)^2}{b^2}=1\), which is option A.

Answer:

4.9) B. \(\frac{(y - k)^2}{a^2}+\frac{(x - h)^2}{b^2}=1\)
4.10) A. \(\frac{(x - h)^2}{a^2}+\frac{(y - k)^2}{b^2}=1\)