Find the standard form of the equation of the ellipse satisfying the given conditions. Foci: (-4,0), (4,0); Vertices: (-5,0) (5,0)

Verified step by step guidance

1. Step 1: Recognize that the foci and vertices are aligned along the x-axis, which means the ellipse is horizontal. The general equation for a horizontal ellipse in standard form is: $x^2/a^2\; +\; y^2/b^2\; =\; 1$.
2. Step 2: Identify the center of the ellipse. The center is the midpoint of the segment connecting the vertices. Since the vertices are (-5, 0) and (5, 0), the center is at (0, 0).
3. Step 3: Determine the value of 'a', the distance from the center to each vertex. The distance from the center (0, 0) to a vertex (5, 0) is 5. Thus, $a\; =\; 5$, and $a^2\; =\; 25$.
4. Step 4: Determine the value of 'c', the distance from the center to each focus. The distance from the center (0, 0) to a focus (4, 0) is 4. Thus, $c\; =\; 4$, and $c^2\; =\; 16$.
5. Step 5: Use the relationship $c^2\; =\; a^2\; -\; b^2$ to find $b^2$. Substituting $c^2\; =\; 16$ and $a^2\; =\; 25$, solve for $b^2$. Once $b^2$ is found, substitute $a^2$ and $b^2$ into the standard form equation.

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 varies based on its orientation, either horizontal or vertical, and is crucial for identifying its properties such as foci, vertices, and axes.

Standard Form of an Ellipse

The standard form of the equation of an ellipse centered at the origin is given by (x²/a²) + (y²/b²) = 1 for a horizontal ellipse, where 'a' is the distance from the center to the vertices along the x-axis, and 'b' is the distance along the y-axis. For a vertical ellipse, the form is (x²/b²) + (y²/a²) = 1. Understanding this form is essential for deriving the equation from given foci and vertices.

Distance Between Foci and Vertices

The distance between the foci and the vertices of an ellipse is related to its semi-major and semi-minor axes. The distance 'c' from the center to each focus is calculated using the formula c² = a² - b², where 'a' is the semi-major axis and 'b' is the semi-minor axis. This relationship is vital for determining the parameters needed to write the standard form of the ellipse's equation.