Graph the ellipse and locate the foci. (y^2)/25 + (x^2)/16 = 1

Verified step by step guidance

1. Rewrite the equation of the ellipse in standard form. The given equation is \( \frac{y^2}{25} + \frac{x^2}{16} = 1 \). Notice that the larger denominator (25) is under \(y^2\), which means the major axis is vertical.
2. Identify the key components of the ellipse. The standard form for a vertical ellipse is \( \frac{y^2}{a^2} + \frac{x^2}{b^2} = 1 \), where \(a^2 = 25\) and \(b^2 = 16\). From this, \(a = 5\) and \(b = 4\).
3. Determine the foci of the ellipse. The distance from the center to each focus is given by \(c = \sqrt{a^2 - b^2}\). Substitute \(a^2 = 25\) and \(b^2 = 16\) into the formula to find \(c\).
4. Locate the foci. Since the major axis is vertical, the foci will be located at \((0, c)\) and \((0, -c)\), where \(c\) is the value calculated in the previous step.
5. Graph the ellipse. Plot the center at \((0, 0)\), draw the vertices at \((0, 5)\) and \((0, -5)\), and the co-vertices at \((4, 0)\) and \((-4, 0)\). Then sketch the ellipse, ensuring it is elongated along the vertical axis.

Key Concepts

Ellipse Definition

An ellipse is a set of points in a plane where the sum of the distances from two fixed points, called foci, is constant. The standard form of an ellipse's equation is (y^2/a^2) + (x^2/b^2) = 1, where 'a' and 'b' are the semi-major and semi-minor axes, respectively. Understanding this definition is crucial for graphing the ellipse and identifying its key features.

Foci of an Ellipse

The foci of an ellipse are two specific points located along the major axis, which play a significant role in its geometric properties. For an ellipse centered at the origin, the foci can be found using the formula c = √(a^2 - b^2), where 'c' is the distance from the center to each focus. Identifying the foci is essential for understanding the ellipse's shape and its reflective properties.

Graphing an Ellipse

Graphing an ellipse involves plotting its center, vertices, and foci based on the values of 'a' and 'b' from its equation. The semi-major axis 'a' determines the vertical or horizontal stretch, while the semi-minor axis 'b' determines the perpendicular stretch. By accurately plotting these points and drawing a smooth curve, one can effectively represent the ellipse on a coordinate plane.