The graph of an ellipse is shown with center at $(2,3)$, a major axis of length $6$ parallel to the x-axis, and a minor axis of length $4$. Which equation represents this ellipse?

• A. $\frac{{(x-2)}^2}{9}+\frac{{(y-3)}^2}{4}=1$
• B. $\frac{{(x-2)}^2}{6}+\frac{{(y-3)}^2}{4}=1$
• C. $\frac{{(x-2)}^2}{4}+\frac{{(y-3)}^2}{9}=1$
• D. $\frac{{(x-2)}^2}{3}+\frac{{(y-3)}^2}{2}=1$

Answer: (A)

Verified step by step guidance

1. Identify the center of the ellipse, which is given as (2, 3). This means the ellipse equation will be centered at these coordinates, so the terms will be (x - 2) and (y - 3).
2. Determine the lengths of the major and minor axes. The major axis length is 6, so the semi-major axis length \( a \) is half of that: \( a = \frac{6}{2} = 3 \). The minor axis length is 4, so the semi-minor axis length \( b \) is \( b = \frac{4}{2} = 2 \).
3. Since the major axis is parallel to the x-axis, \( a \) corresponds to the x-direction and \( b \) corresponds to the y-direction. The standard form of the ellipse equation centered at (h, k) with major axis along the x-axis is: \[\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\]
4. Substitute the center (h, k) = (2, 3), \( a = 3 \), and \( b = 2 \) into the equation: \[\frac{(x - 2)^2}{3^2} + \frac{(y - 3)^2}{2^2} = 1\]
5. Simplify the denominators to get the final form of the ellipse equation: \[\frac{(x - 2)^2}{9} + \frac{(y - 3)^2}{4} = 1\]