Problem 28 - A Graphical Approach to College Algebra

Question

Determine the type of conic section represented by each equation, and graph it, provided a graph exists. $$9 x^{2}+36 y^{2}=36$$

Step-by-Step Solution

Answer

The equation represents an ellipse.

Step 1: Identify the Standard Form

The standard form of the equation for an ellipse is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Compare this with the given equation \(9x^2 + 36y^2 = 36\).

Step 2: Simplify the Original Equation

Divide all the terms of the equation \(9x^2 + 36y^2 = 36\) by 36 to simplify it. This gives \(\frac{9x^2}{36} + \frac{36y^2}{36} = 1\).

Step 3: Simplify Numerator and Denominator

Simplify each term: \(\frac{9x^2}{36} = \frac{x^2}{4}\) and \(\frac{36y^2}{36} = \frac{y^2}{1}\). This results in the equation \(\frac{x^2}{4} + \frac{y^2}{1} = 1\).

Step 4: Determine the Type of Conic Section

The equation \(\frac{x^2}{4} + \frac{y^2}{1} = 1\) is in the standard form of an ellipse where \(a^2 = 4\) and \(b^2 = 1\). Therefore, the conic section is an ellipse.

Key Concepts

Ellipses

An ellipse is one of the fascinating shapes in geometry known as conic sections. Imagine a stretched circle — that's an ellipse! 🌟 It's kind of like squashing or elongating a circle along one axis. Ellipses are defined by **two axes**:

• **Major Axis**: The longer axis of the ellipse.
• **Minor Axis**: The shorter one.

These axes help determine the shape and orientation of the ellipse. What's unique and important about ellipses is that they always have a sum of distances from any point on the ellipse to two fixed points, called foci, which remains constant. Typical real-world examples include planetary orbits around the sun which are elliptical.
Ellipses can look pretty, but they are also a part of our day-to-day world. Understanding how to manipulate their equations can help you visualize and solve problems linked to them.

Standard Form of Conic Sections

Understanding the standard form of conic section equations is crucial in determining shapes like ellipses, circles, hyperbolas, and parabolas. Each conic section has its own equation pattern.
For ellipses, the **standard form** is expressed as: \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]where:

• \(a^2\) and \(b^2\) are the squared lengths of the semi-major and semi-minor axes, respectively.
• If \(a^2 > b^2\), the major axis is horizontal.
• If \(b^2 > a^2\), the major axis is vertical.

This form makes it straightforward to identify the shape of the ellipse and calculate its properties.
One of the neat tricks is rewriting the given equation in this standard form. This helps to identify and differentiate between different conic sections, particularly when graphing or analyzing them geometrically.

Equation Simplification

Equation simplification is like organizing a messy room — it makes everything clearer and easier to find. Simplifying an equation, especially for conic sections, involves restructuring it into a more recognizable and standard form.
Take the given equation: \[9x^2 + 36y^2 = 36\]To simplify, divide every term by 36 to reduce it to:\[\frac{x^2}{4} + \frac{y^2}{1} = 1\]This resembles the standard form, which is essential for **recognizing the shape** and **making calculations easy**. In this case, \[\frac{x^2}{4} + \frac{y^2}{1} = 1\]confirms that it’s an ellipse.
Instead of battling with a complicated equation, these simplification steps help you view the math in a clearer, simpler form. Simplification isn’t just about making an equation look nicer — it is a fundamental process to decode and understand the mathematical relationships it represents.