Graph each ellipse and give the location of its foci. x²/25 + (y -2)² /36= 1

Verified step by step guidance

1. Identify the standard form of the ellipse equation: \(\frac{x^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\), where \((h, k)\) is the center of the ellipse. In this equation, \(h = 0\) and \(k = 3\).
2. Determine the values of \(a^2\) and \(b^2\) by comparing the denominators: \(a^2 = 9\) and \(b^2 = 49\). Since \(b^2 > a^2\), the major axis is vertical.
3. Find the lengths of the semi-major axis \(b = \sqrt{49} = 7\) and the semi-minor axis \(a = \sqrt{9} = 3\).
4. Locate the center of the ellipse at the point \((0, 3)\).
5. Calculate the distance \(c\) from the center to each focus using the formula \(c = \sqrt{b^2 - a^2} = \sqrt{49 - 9}\). The foci will be located along the major axis (vertical) at \((0, 3 \pm c)\).

Key Concepts

Standard Form of an Ellipse

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

Identifying the Major and Minor Axes

The larger denominator between a² and b² determines the length of the major axis, while the smaller corresponds to the minor axis. The major axis is the longest diameter of the ellipse, and its orientation (horizontal or vertical) depends on whether a² or b² is larger. This distinction is crucial for graphing and locating the foci.

Finding the Foci of an Ellipse

The foci are two fixed points inside the ellipse, located along the major axis, defined by the distance c from the center, where c² = |a² - b²|. Knowing c allows you to find the exact coordinates of the foci, which are essential for understanding the ellipse's geometric properties and for accurate graphing.