Problem 8 - Algebra 2

Question

Write an equation of an ellipse in standard form with center at the origin and with the given vertex and co-vertex. $$ (0,5),(-3,0) $$

Step-by-Step Solution

Answer

The equation of the ellipse in standard form is \(\frac{x^2}{25} + \frac{y^2}{9} = 1\)

Step 1: Identify the semi-major axis (a) and the semi-minor axis (b).

The vertex represents the semi-major axis of the ellipse. It is located at (0,5), implying that the value of a is 5. Similarly, the co-vertex is the semi-minor axis of the ellipse. It is located at (-3,0), which means the value of b is 3. So, a = 5 and b= 3.

Step 2: Write the standard form of the ellipse.

The standard form of the ellipse with center at the origin (h, k) = (0, 0) is given by \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where a is the length of the semi-major axis, and b is the length of the semi-minor axis.

Step 3: Substitute a and b into the equation.

Substituting a = 5 and b = 3 into the standard ellipse equation: \(\frac{x^2}{5^2} + \frac{y^2}{3^2} = 1\)

Key Concepts

Standard Form of an Ellipse

The standard form of an ellipse is a way to express the equation of an ellipse that is centered at the origin. This is particularly useful in simplifying the graphing of ellipses and in understanding their geometric properties. The equation of an ellipse with its center at the origin

• The general form is given by \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\),
• where \(a\) represents the semi-major axis, and \(b\) represents the semi-minor axis.

The main goal of using the standard form is to simplify the representation of the ellipse by having all the important parameters easily identifiable, which results in easier computations. To convert any set of ellipse data into standard form, determine your \(a\) and \(b\) values, then substitute these into the equation.

Semi-Major Axis

In an ellipse, the semi-major axis is the longest radius extending from the center to the boundary in the direction of the longest axis. It's often denoted as \(a\) in the equation of an ellipse. Knowing the semi-major axis is crucial because:

• It defines half of the total extent of the ellipse across its longest dimension.
• The length of \(a\) helps to define the elliptical shape. A larger \(a\) relative to \(b\) makes the ellipse more stretched horizontally or vertically.

In our specific case, the vertex at (0,5) indicates that the semi-major axis is vertical, hence, \(a = 5\). This means that from the center of the ellipse at the origin (0,0), the ellipse stretches vertically 5 units up and down.

Semi-Minor Axis

The semi-minor axis is the shortest radius in an ellipse, extending from the center to the ellipse’s edge, perpendicular to the semi-major axis. It is often denoted as \(b\) in the standard form of an ellipse. Understanding the semi-minor axis involves:

• Recognizing that it accounts for half of the width of the ellipse across its shortest dimension.
• A smaller \(b\), as compared to \(a\), results in a more elongated ellipse.

In the exercise example, the co-vertex at (-3,0) aligns with the x-axis, indicating that the semi-minor axis is horizontal. Here, \(b = 3\). This means the ellipse stretches 3 units to the left and right from the center at the origin.