Problem
Find the expanded form of an ellipse with a center at (3, -2), a horizontal semi-axis of 5 units, and a vertical semi-axis of 3 units.
Solution
Step 1 ：The general form of the equation of an ellipse is \(\frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1\), where (h, k) is the center of the ellipse, a is the semi-major axis, and b is the semi-minor axis.
Step 2 ：Substitute the given values into the equation: h = 3, k = -2, a = 5, and b = 3.
Step 3 ：After substituting, we get \(\frac{(x-3)^{2}}{5^{2}} + \frac{(y+2)^{2}}{3^{2}} = 1\).
