Given an ellipse centered at the origin with a major axis along the x-axis, a length of 10 for the major axis, and a length of 6 for the minor axis, which equation represents this ellipse?

• A. $\frac{x^2}{9}+\frac{y^2}{25}=1$
• B. $\frac{x^2}{36}+\frac{y^2}{100}=1$
• C. $\frac{x^2}{25}+\frac{y^2}{9}=1$
• D. $\frac{x^2}{16}+\frac{y^2}{25}=1$

Answer: (C)

Verified step by step guidance

1. Recall the standard form of the equation of an ellipse centered at the origin with the major axis along the x-axis: \(\frac{{x^2}}{{a^2}} + \frac{{y^2}}{{b^2}} = 1\), where \(a\) is the semi-major axis length and \(b\) is the semi-minor axis length.
2. Identify the lengths of the major and minor axes given: the major axis length is 10, so the semi-major axis \(a\) is half of that, which is 5; the minor axis length is 6, so the semi-minor axis \(b\) is half of that, which is 3.
3. Substitute the values of \(a\) and \(b\) into the ellipse equation: replace \(a\) with 5 and \(b\) with 3, giving \(\frac{{x^2}}{{5^2}} + \frac{{y^2}}{{3^2}} = 1\).
4. Simplify the denominators by squaring the semi-axis lengths: \$5^2 = 25\( and \)3^2 = 9$, so the equation becomes \(\frac{{x^2}}{{25}} + \frac{{y^2}}{{9}} = 1\).
5. Compare this equation with the given options to identify the correct one that matches \(\frac{{x^2}}{{25}} + \frac{{y^2}}{{9}} = 1\).