$$ \frac{\left(x-1\right)^2}{3^2} $$+$$ \frac{\left(y-5\right)^2}{4^2} $$=1

The equation you provided is:

\[
\frac{(x - 1)^2}{3^2} + \frac{(y - 5)^2}{4^2} = 1
\]

This is the standard form of an ellipse equation:

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

where:
- $(h, k)$ is the center of the ellipse,
- $a$ is the semi-major axis (horizontal radius),
- $b$ is the semi-minor axis (vertical radius).

### Key Information:
- The center of the ellipse is at $(h, k) = (1, 5)$.
- The semi-major axis is 4, because $b = 4$.
- The semi-minor axis is 3, because $a = 3$.
- The major axis is vertical because $b > a$.

### Conclusion:
This ellipse is centered at $(1, 5)$ with a vertical semi-major axis of 4 units and a horizontal semi-minor axis of 3 units.

Would you like more details or have any questions?

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Here are 5 related questions to explore further:
1. How do you calculate the foci of an ellipse?
2. What is the general equation for ellipses not centered at the origin?
3. How does the orientation of the ellipse change when $a > b$?
4. How do we graph an ellipse using its standard form equation?
5. How is an ellipse related to a circle in conic sections?

**Tip:** The foci of an ellipse can be found using the formula $c = \sqrt{b^2 - a^2}$ where $c$ is the distance from the center to each focus.

Explore the equation of an ellipse (x - 1)^2/9 + (y - 5)^2/16 = 1 and learn how to determine its center, semi-major axis, semi-minor axis, and orientation. Discover key mathematical concepts in conic sections, perfect for students in Grades 10-12.

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