Problem 30 - Algebra 2

Question

Write an equation of an ellipse for the given foci and co-vertices. foci \((0, \pm 4),\) co-vertices \(( \pm 2,0)\)

Step-by-Step Solution

Answer

The equation of the ellipse is \(\frac{x^2}{4} + \frac{y^2}{20} = 1\).

Step 1: Identify given information

The given information is that the foci are at (0,±4) and the co-vertices are at (±2,0). This means c=4 and b=2.

Step 2: Calculate the semi-major axis

Use the equation \(a^2 = b^2 + c^2\) to calculate the semi-major axis. In this case, \(a^2 = 2^2 + 4^2 = 20\), hence a is the square root of 20 or in simplified form \(2\sqrt{5}\).

Step 3: Construct the equation of the ellipse

The equation of the ellipse when foci are on y-axis and the centre is at the origin is given by \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\). Substituting b=2 and a=\(2\sqrt{5}\) we get the equation \(\frac{x^2}{4} + \frac{y^2}{20} = 1\).

Key Concepts

Foci of an Ellipse

The foci of an ellipse are key in defining its shape and position. They are two distinct points located along the major axis. For the ellipse given in the exercise, the foci are at the coordinates

• (0, 4) and (0, -4)

This indicates that the foci are positioned vertically along the y-axis. The distance from the center of the ellipse (which is at the origin

• (0, 0)

) to each focus is known as the focal distance "c." In this specific example,

• c = 4

The foci play a critical role in the equation of an ellipse and in various applications of ellipse geometry. Understanding the role of foci helps visualize the stretch and size of the ellipse along its major axis.

Co-vertices and Their Role

Co-vertices are another important aspect of ellipse geometry. These points lie on the ellipse's minor axis, perpendicular to the major axis. For the exercise at hand, the co-vertices are located at

• (±2, 0)

This arrangement indicates that the minor axis is aligned with the x-axis. The distance from the center to each co-vertex is labelled "b." Here,

• b = 2

Co-vertices help define the shorter "span" of the ellipse, or its minor diameter. Grasping the concept of co-vertices is essential as it helps in understanding how the ellipse is stretched along its minor axis, which complements the major axis defined by the foci.

Understanding the Semi-Major Axis

The semi-major axis of an ellipse is the longest radius extending from the center to the edge along the major axis. In this exercise, the semi-major axis is calculated using the formula:

• \[ a^2 = b^2 + c^2 \]

Given that

• b = 2
• c = 4

we find:

• \[ a^2 = 2^2 + 4^2 = 20 \]

This yields a = \(2\sqrt{5}\). The semi-major axis stretches from the center of the ellipse to its furthest point along the major axis, guiding the ellipse's overall size and shape.

Exploring Ellipse Geometry

Ellipse geometry is a fascinating area of mathematics, heavily reliant on understanding its axes, foci, and vertices. The standard equation of an ellipse with a vertical major axis centered at the origin is

• \[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \]

For this exercise:

• a = 2\sqrt{5}
• b = 2

Substituting these values into the standard form, the equation becomes

• \[ \frac{x^2}{4} + \frac{y^2}{20} = 1 \]

This equation fully characterizes the ellipse's shape and size, highlighting the balance between its longer and shorter dimensions, aligned with its foci and co-vertices.