Problem 28 - Algebra 2

Question

Write an equation of an ellipse for the given foci and co-vertices. foci \((0, \pm 8),\) co-vertices \(( \pm 8,0)\)

Step-by-Step Solution

Answer

The equation of the ellipse given the foci at (0,±8) and co-vertices at (±8,0) is \(\frac{x^2}{128} + \frac{y^2}{64} = 1\).

Step 1: Identify the coordinates of foci and co-vertices

The foci are at coordinates (0,±8) and the co-vertices are at (±8,0). This suggests that the center of the ellipse is at the origin (0,0).

Step 2: Calculate the distances 'b' and 'c'

The distance 'c' from the center to each focus is 8 (the y-coordinate of the foci) and the distance 'b' to each co-vertex is also 8 (the x-coordinate of the co-vertices).

Step 3: Compute for 'a'

By definition, in an ellipse, \(a^2 = b^2 + c^2\). Substituting the values of 'b' and 'c' into the equation: \(a^2 = 8^2 + 8^2 = 128\). So, 'a' is \( \sqrt{128} = 8\sqrt{2}\).

Step 4: Write the equation of the ellipse

The standard equation of an ellipse with center at the origin in the coordinate plane is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Substituting our values for 'a' and 'b', the equation of the ellipse is \(\frac{x^2}{128} + \frac{y^2}{64} = 1\).

Key Concepts

Foci of Ellipse

The foci of an ellipse are two points located inside the ellipse. They play a crucial role in its geometric definition. Let's explore what makes them special. The foci are a fundamental aspect because the sum of the distances from any point on the ellipse to each focus is constant. This is the key defining property of an ellipse. In our given example, the foci are at

• (0, 8)
• (0, -8)

This means they lie on the y-axis. The distance of each focus from the center of the ellipse, called 'c', is 8 units. The foci help establish the orientation and dimensions of the ellipse, influencing its shape by dictating the positions that are closest and furthest from the center.

Co-Vertices of Ellipse

Co-vertices are another important aspect when discussing ellipses. These are the points where the ellipse intersects its minor axis. The ellipse is symmetrical about both the x-axis and y-axis, as illustrated by its co-vertices

• (8, 0)
• (-8, 0)

In this example, the co-vertices are situated along the x-axis, meaning the minor axis is horizontal. The distance from the center to a co-vertex is labeled 'b'. Here, 'b' equals 8 units. This distance helps define the width of the ellipse across its minor axis, demonstrating that the closer the co-vertices are to the center, the narrower the ellipse is.

Standard Form of Ellipse Equation

The equation of an ellipse in its standard form helps us represent the curve algebraically. The standard form of an ellipse centered at the origin \[ (0, 0) \] is \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] Here, 'a' represents the semi-major axis' length whereas 'b' is the length of the semi-minor axis. These axes are the longest and shortest diameters respectively. Calculating 'a' and 'b' is essential for setting up this equation. Given 'b'=8 and 'c'=8, we calculated 'a' using \[ a^2 = b^2 + c^2 \] resulting in \[ a^2 = 128 \] and thus \[ a = 8\sqrt{2} \]. Thus, the equation \[ \frac{x^2}{128} + \frac{y^2}{64} = 1 \] describes our ellipse. Understanding this formula allows us to graph and analyze ellipses in various contexts more effectively.