https://r2.erweima.ai/mathbot/image/172643320582458fevovq.jpg

The equation provided in the image is:

\[
\frac{(x - 2)^2}{16} + \frac{(y - 1)^2}{25} = 1
\]

This is the standard form of the equation of an ellipse, given by:

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

Where:
- \((h, k)\) is the center of the ellipse.
- \(a^2\) and \(b^2\) are the squares of the lengths of the semi-major and semi-minor axes.

In this case:
- The center of the ellipse is \((2, 1)\).
- The denominators \(a^2 = 25\) and \(b^2 = 16\), meaning \(a = 5\) and \(b = 4\).
- The value of \(a = 5\) is greater than \(b = 4\), so the major axis is vertical, and the minor axis is horizontal.

Now let's address each of the statements:

**a) La coordenada del centro de la elipse es (2,1):**  
- This is **true** because the center is \((2, 1)\), matching the standard form.

**b) La longitud del eje mayor de la elipse es 5:**  
- This is **false**. The length of the major axis is \(2a\), where \(a = 5\). Thus, the total length of the major axis is \(2 \times 5 = 10\).

**c) La elipse tiene un eje focal vertical:**  
- This is **true** because the larger value is under \((y - 1)^2\), indicating that the major axis (and thus the focal axis) is vertical.

Let me know if you want more details!

### Related Questions:
1. What is the length of the minor axis of this ellipse?
2. How do you find the foci of an ellipse?
3. What is the equation of an ellipse if the major axis is horizontal?
4. How can you determine whether an ellipse's major axis is vertical or horizontal?
5. What is the distance between the foci of this ellipse?

**Tip:** Always check the denominators in an ellipse's equation to find the lengths of the axes!

Dada la siguiente ecuación de la elipse: (x - 2)^2 / 16 + (y - 1)^2 / 25 = 1. Determine if the following statements are true or false: a) La coordenada del centro de la elipse es (2,1), b) La longitud del eje mayor de la elipse es 5, c) La elipse tiene un eje focal vertical.

Learn how to find the center, length of the major axis, and orientation of an ellipse from its equation. This problem focuses on the equation of an ellipse, covering key mathematical concepts such as conic sections, and is suitable for students in Grades 9-12.

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