Graph each ellipse and give the location of its foci. (x − 1)²/2 + (y +3)² /5= 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 \(\frac{(x - 2)^2}{7} + \frac{(y + 1)^2}{11} = 1\), determine the center of the ellipse as \((2, -1)\).
3. Compare the denominators to find \(a^2\) and \(b^2\). Since \$11 > 7\(, set \)a^2 = 11\( and \)b^2 = 7$. This means the major axis is vertical.
4. Calculate the distance \(c\) from the center to each focus using the formula \(c = \sqrt{a^2 - b^2}\). Substitute the values to find \(c = \sqrt{11 - 7}\).
5. Locate the foci along the major axis (vertical axis) by adding and subtracting \(c\) from the \(y\)-coordinate of the center. The foci are at \((2, -1 + c)\) and \((2, -1 - c)\).

Key Concepts

Standard Form of an Ellipse

An ellipse can be expressed in the standard form \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), where \((h, k)\) is the center. The denominators \(a^2\) and \(b^2\) represent the squares of the lengths of the semi-major and semi-minor axes, respectively. Identifying these values helps in graphing the ellipse accurately.

Determining the Orientation of the Ellipse

The larger denominator between \(a^2\) and \(b^2\) indicates the major axis direction: if \(a^2 > b^2\), the ellipse is stretched horizontally; if \(b^2 > a^2\), it is stretched vertically. This orientation is crucial for locating the foci and sketching the ellipse.

Finding the Foci of an Ellipse

The foci lie along the major axis, located at a distance \(c\) from the center, where \(c = \sqrt{|a^2 - b^2|}\). Knowing \(c\) and the center coordinates allows you to find the exact positions of the foci, which are key points defining the ellipse's shape.