Problem 29 - Algebra 2

Question

Write an equation for the ellipse that satisfies each set of conditions. Write the equation \(10 x^{2}+2 y^{2}=40\) in standard form.

Step-by-Step Solution

Answer

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

Step 1: Identify the original equation

The given equation of the ellipse is \(10x^2 + 2y^2 = 40\). This is not in the standard form of an ellipse equation, which is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\). We need to rearrange and simplify the given equation to match this standard form.

Step 2: Simplify the equation

First, divide every term in the equation \(10x^2 + 2y^2 = 40\) by 40. This simplifies to \(\frac{10x^2}{40} + \frac{2y^2}{40} = 1\).

Step 3: Simplify fractions

Let's reduce the fractions: \(\frac{10}{40} = \frac{1}{4}\) and \(\frac{2}{40} = \frac{1}{20}\). So, the equation becomes \(\frac{x^2}{4} + \frac{y^2}{20} = 1\).

Step 4: Identify the standard form parameters

Now the equation is in the standard form of an ellipse, where \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), with \(h = 0\), \(k = 0\), \(a^2 = 4\), and \(b^2 = 20\). Therefore, \(a = 2\) and \(b = \sqrt{20} = 2\sqrt{5}\).

Key Concepts

Standard Form of Ellipse

When discussing ellipses in mathematics, it's crucial to understand the standard form of their equations. The standard form of an ellipse helps in easily identifying the key properties of the ellipse, such as its center, axes lengths, and orientation. For an ellipse centered at the origin, the equation typically looks like this:
\[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]
Here,

• \( (h, k) \) represents the center of the ellipse. In our example, this is \( (0,0) \), indicating a central origin.
• \( a \) and \( b \) are the semi-major and semi-minor axes, respectively. They dictate the length of the ellipse along its axes.

Knowing this form makes it easier to visualize and analyze the ellipse's size and orientation.

Converting Equations

Converting an equation into the standard form involves rearranging and simplifying. Let's walk through the process using our example.
The initial equation is given as\( 10x^2 + 2y^2 = 40 \). This format is not immediately helpful for identifying the ellipse properties. By rewriting it to match the standard form, we can better understand the ellipse it represents.
Start by dividing each term by the constant on the right side, here it's 40. This normalizes the equation to have 1 on the right-hand side, which is key for standard form:\[ \frac{10x^2}{40} + \frac{2y^2}{40} = 1 \]
Now, the equation is more structured and reflects the ellipse's characteristic form, allowing further simplification.

Simplifying Fractions

Simplifying fractions is a fundamental step in converting any equation to its simplest form, especially when dealing with ellipses. For our specific example:

• First, simplify the fraction \( \frac{10x^2}{40} \) to \( \frac{1}{4}x^2 \). This is done by dividing the numerator and denominator by their greatest common divisor.
• Similarly, simplify \( \frac{2y^2}{40} \) to \( \frac{1}{20}y^2 \).

These simplifications enable the transformed equation to be written as:\[ \frac{x^2}{4} + \frac{y^2}{20} = 1 \]
By reducing fractions, the equation becomes cleaner and easier to work with, facilitating further analysis of the ellipse.

Parameters of Ellipse

Identifying the parameters of an ellipse is crucial for understanding its geometry. Once in standard form, as in our example \( \frac{x^2}{4} + \frac{y^2}{20} = 1 \), these parameters are clear.

• \( h \) and \( k \) define the center of the ellipse. In this equation, they are both 0, indicating the ellipse is centered at the origin.
• \( a^2 = 4 \), so \( a = 2 \). This means the semi-major axis (along the x-direction here) is 2 units.
• \( b^2 = 20 \), leading to \( b = \sqrt{20} = 2\sqrt{5} \). This is the length of the semi-minor axis along the y-direction.

These parameters inform not only the size of the ellipse but also help in sketching its position in a coordinate plane. Understanding these elements makes the study of ellipses highly practical in both mathematics and science.