Graph each ellipse and give the location of its foci. (x +3)²+ 4(y -2)² = 16

Verified step by step guidance

1. Identify the standard form of the ellipse equation. The given equation is \( (x + 1)^2 + 4(y + 4)^2 = 100 \). To write it in the standard form of an ellipse, divide both sides by 100 to get \( \frac{(x + 1)^2}{100} + \frac{4(y + 4)^2}{100} = 1 \).
2. Simplify the second term by dividing 4 by 100: \( \frac{(x + 1)^2}{100} + \frac{(y + 4)^2}{25} = 1 \). Now the equation is in the form \( \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \), where the center is at \((-1, -4)\).
3. Determine the values of \(a^2\) and \(b^2\). Here, \(a^2 = 100\) and \(b^2 = 25\). Since \(a^2 > b^2\), the major axis is horizontal, and the ellipse stretches more along the x-axis.
4. Find the lengths of the semi-major axis \(a\) and semi-minor axis \(b\) by taking the square roots: \(a = \sqrt{100} = 10\) and \(b = \sqrt{25} = 5\).
5. Calculate the focal distance \(c\) using the relationship \(c^2 = a^2 - b^2\). Then, find the coordinates of the foci by moving \(c\) units left and right from the center along the major axis (x-axis). The foci are at \((-1 - c, -4)\) and \((-1 + c, -4)\).

Key Concepts

Standard Form of an Ellipse

The standard form of an ellipse equation is (x-h)²/a² + (y-k)²/b² = 1 or (x-h)²/b² + (y-k)²/a² = 1, where (h, k) is the center. It shows how the ellipse is stretched along the x- and y-axes. Recognizing this form helps identify the ellipse's center and the lengths of its axes.

Identifying the Center and Axes Lengths

From the equation, the center is at (-1, -4) by reversing the signs inside the parentheses. The denominators under the squared terms determine the lengths of the semi-major and semi-minor axes. Dividing both sides by 100 puts the equation in standard form, revealing these lengths.

Finding the Foci of an Ellipse

The foci lie along the major axis, located at a distance c from the center, where c² = a² - b². Here, a is the semi-major axis length and b the semi-minor axis length. Calculating c allows you to find the exact coordinates of the foci, essential for graphing the ellipse accurately.