To find the standard form of the equation of an ellipse, we use the formula:

$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

where:

• $(h, k)$ is the center of the ellipse,
• $a$ is the semi-major axis length,
• $b$ is the semi-minor axis length.

Given:

• Center $(h, k) = (5, 3)$,
• Horizontal major axis length = 20, so $a = \frac{20}{2} = 10$,
• Minor axis length = 16, so $b = \frac{16}{2} = 8$.

Step 1: Identify the values.

• $h = 5$
• $k = 3$
• $a = 10$
• $b = 8$

Step 2: Plug these values into the standard form.

Since the major axis is horizontal, $a^2$ will be under the $(x - h)^2$ term.

$\frac{(x - 5)^2}{10^2} + \frac{(y - 3)^2}{8^2} = 1$

Simplifying the equation:

$\frac{(x - 5)^2}{100} + \frac{(y - 3)^2}{64} = 1$

Final Answer: Standard Form

The standard equation of the ellipse is:

$\frac{(x - 5)^2}{100} + \frac{(y - 3)^2}{64} = 1$

This represents an ellipse centered at $(5, 3)$, with a horizontal major axis of length 20 and a vertical minor axis of length 16.

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Would you like to go over the details, or do you have any questions? Here are some related questions to explore:

1. How do you distinguish between the major and minor axes in an ellipse equation?
2. What is the significance of the values $a^2$ and $b^2$ in the ellipse's equation?
3. How would the equation change if the major axis were vertical instead of horizontal?
4. How do you find the foci of an ellipse given its equation?
5. How do the equations of ellipses differ from those of circles?

Tip: Remember that the larger denominator in the ellipse's equation corresponds to the square of the semi-major axis, which is always associated with the longer axis of the ellipse.