Graph each ellipse and give the location of its foci. (x − 4)²/9 + (y +2)² /25= 1

Verified step by step guidance

1. Identify the standard form of the ellipse equation: \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\), where \((h, k)\) is the center of the ellipse.
2. From the given equation \(\frac{(x - 3)^2}{4} + \frac{(y + 1)^2}{16} = 1\), determine the center as \((3, -1)\).
3. Compare the denominators to find \(a^2\) and \(b^2\). Here, \(a^2 = 16\) and \(b^2 = 4\). Since \(a^2 > b^2\), the major axis is vertical.
4. Calculate the distance \(c\) from the center to each focus using the formula \(c = \sqrt{a^2 - b^2}\).
5. Locate the foci at \((h, k \pm c)\) because the major axis is vertical, so the foci lie above and below the center along the y-axis.

Key Concepts

Standard Form of an Ellipse

The standard form of an ellipse equation is (x - h)²/a² + (y - k)²/b² = 1, where (h, k) is the center. The values a² and b² represent the squares of the lengths of the semi-major and semi-minor axes. Understanding this form helps in identifying the ellipse's size, shape, and position on the coordinate plane.

Major and Minor Axes

The major axis is the longest diameter of the ellipse, while the minor axis is the shortest. The larger denominator (a² or b²) corresponds to the major axis. Knowing which axis is major or minor is essential for graphing the ellipse accurately and determining the orientation (horizontal or vertical).

Foci of an Ellipse

The foci are two fixed points inside the ellipse such that the sum of distances from any point on the ellipse to the foci is constant. Their locations are found using c² = |a² - b²|, where c is the distance from the center to each focus along the major axis. Identifying the foci is crucial for understanding the ellipse's geometric properties.