Let's analyze the given ellipse:

Step 1: Identify the center

From the graph, we can see that the center of the ellipse is at (0,0).

Step 2: Identify the lengths of the axes

• The major axis is vertical (along the y-axis).
• The minor axis is horizontal (along the x-axis).

By observing the graph:

• The semi-major axis (longest radius) extends from $y = 0$ to $y = \pm 5$, so $a = 5$.
• The semi-minor axis (shortest radius) extends from $x = 0$ to $x = \pm 3$, so $b = 3$.

Step 3: Write the standard equation

Since the major axis is vertical, the equation of the ellipse is:

$\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$

Substituting $a = 5$ and $b = 3$:

$\frac{x^2}{3^2} + \frac{y^2}{5^2} = 1$

$\frac{x^2}{9} + \frac{y^2}{25} = 1$

Step 4: Find the foci

The formula for the focal distance $c$ in an ellipse is:

$c^2 = a^2 - b^2$

$c^2 = 5^2 - 3^2 = 25 - 9 = 16$

$c = \sqrt{16} = 4$

Since the foci are along the major axis (y-axis), their coordinates are:

$(0, \pm c) = (0, \pm 4)$

Final Answer:

• Equation of the ellipse: $\frac{x^2}{9} + \frac{y^2}{25} = 1$

• Foci: $(0,4)$ and $(0,-4)$

Would you like me to provide a graphical representation or a deeper explanation of the steps?

Related Questions:

1. How do you determine whether an ellipse is vertical or horizontal?
2. What is the relationship between the foci and the eccentricity of an ellipse?
3. How do you find the equation of an ellipse given its foci and vertices?
4. What are the real-life applications of ellipses in physics and engineering?
5. How would the equation change if the center of the ellipse was shifted to (h,k)?

Tip:

For any ellipse, remember that the foci always lie along the major axis, and their distance from the center is determined by $c^2 = a^2 - b^2$.