To solve this problem, let's analyze the ellipse shown in the image.

Step-by-Step Solution

1. Identify the Center of the Ellipse: The center is given as $(2, -3)$.

2. Determine the Orientation and Lengths of the Axes: By examining the graph, the ellipse is wider horizontally than vertically. This indicates a horizontal orientation.

   • The semi-major axis extends horizontally, and by counting the grid units from the center to the ellipse edge along the x-axis, we find it reaches 4 units to either side. Thus, $a = 4$.
   • The semi-minor axis extends vertically, and by counting grid units, we find it reaches 3 units up and down. So, $b = 3$.

3. Write the Standard Form of the Ellipse Equation: For an ellipse centered at $(h, k)$ with a horizontal orientation, the standard form is: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$ Substituting $h = 2$, $k = -3$, $a = 4$, and $b = 3$: $\frac{(x - 2)^2}{4^2} + \frac{(y + 3)^2}{3^2} = 1$ Simplifying, we get: $\frac{(x - 2)^2}{16} + \frac{(y + 3)^2}{9} = 1$

4. Find the Location of the Foci: For an ellipse, the distance from the center to each focus, $c$, is calculated using $c = \sqrt{a^2 - b^2}$.

   • Substituting $a = 4$ and $b = 3$: $c = \sqrt{4^2 - 3^2} = \sqrt{16 - 9} = \sqrt{7}$
   • Since the ellipse is horizontally oriented, the foci are located $c = \sqrt{7}$ units left and right from the center $(2, -3)$.
     • Right focus: $(2 + \sqrt{7}, -3)$
     • Left focus: $(2 - \sqrt{7}, -3)$

Final Answer

• Standard Form of the Equation: $\frac{(x - 2)^2}{16} + \frac{(y + 3)^2}{9} = 1$

• Locations of the Foci: $(2 + \sqrt{7}, -3) \text{ and } (2 - \sqrt{7}, -3)$

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Related Questions

1. How do you find the lengths of the axes in an ellipse given its equation?
2. How can you determine the orientation of an ellipse from its standard form equation?
3. What are the differences in equations for ellipses oriented horizontally versus vertically?
4. How is the distance between the foci related to the lengths of the ellipse's axes?
5. What happens to the foci if the lengths of the major and minor axes are equal?

Tip

Remember, in an ellipse, $a$ is always the larger of the two distances between the center and the edge along the major axis, while $b$ is for the minor axis.