Problem 73 - Precalculus with Limits

Question

TRUE OR FALSE? In Exercises 73 and 74, determine whether the statement is true or false. Justify your answer. The graph of \(x^2+4y^4-4=0\) is an ellipse.

Step-by-Step Solution

Answer

FALSE, the graph of the given equation is not an ellipse.

Step 1: Convert into correct form

Convert the given equation into the form similar to the standard ellipse equation by making one side of the equation equal to zero. This results in \(x^2+4y^4=4\).

Step 2: Simplify the equation

Simplify the equation by dividing the entire equation by 4, to make it look closer to the standard form of ellipse. This gives the equation \(\frac{x^2}{4}+y^4=1\).

Step 3: Compare with the formula of an Ellipse

If we compare this equation with the standard equation for an ellipse, which is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), we can see that for the equation to be that of an ellipse, the exponent of the \(y\) term must be 2, not 4.

Step 4: Arrive at the conclusion

Since the provided equation doesn't match with the standard ellipse equation, the given statement is FALSE. Therefore, the graph of given equation will not represent an ellipse

Key Concepts

Equation of an Ellipse

When we talk about the equation of an ellipse, we refer to a specific algebraic form that helps us define ellipses in a coordinate plane. An ellipse is a set of points where the sum of the distances from any point on the ellipse to two fixed points (called foci) is constant. In mathematical terms, this takes the shape of a typical equation:

• Horizontal ellipse: \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\)
• Vertical ellipse: \(\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1\)

Here, \((h, k)\) is the center of the ellipse, and \(a\) and \(b\) are the distances from the center to the ends of the major and minor axes respectively. Generally, \(a > b\). If we compare these equations to other conic sections, such as circles or hyperbolas, it's crucial to recognize these unique characteristics of ellipses.

When given any quadratic equation, our first task is often to rearrange and adjust it to fit into one of these standard forms. This process helps us verify whether it truly represents an ellipse.

Standard Form of an Ellipse

The standard form of an ellipse is essential because it gives ellipses their distinctive shapes and properties. A challenge in mathematics is determining if a given equation can represent an ellipse. This task often revolves around converting and comparing equations to their standard forms.

The usual standard form equations are:

• \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\) \( (a > b) \)
• \(\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1\) \( (a < b) \)

Converting a general quadratic equation to this form usually involves completing squares and adjusting the equation so it equals one on the right side. However, not all equations that look similar to these can be turned into an ellipse's standard form.

In the case of the equation \(x^2 + 4y^4 = 4\), when you divide by 4 across the board, you get \(\frac{x^2}{4} + y^4 = 1\), but the \(y\) term's exponent isn't the required 2. Therefore, this particular equation cannot be made to fit the standard ellipse form, proving it not to be an ellipse.

Graphing Conics

Graphing conic sections like ellipses involves understanding their algebraic representation and translating it visually onto a plane. Conics include circles, ellipses, parabolas, and hyperbolas, each distinguished by their specific forms and characteristics.

For an ellipse, once in the standard form, graphing is relatively straightforward:

• Identify the center \((h, k)\) from the equation.
• Determine \(a\) and \(b\), which influence the stretch in the horizontal and vertical directions.
• Use \(a\) to plot points along the major axis and \(b\) along the minor axis, extending from the center.

Ellipses are symmetrical, so once you have these key points, sketching the curve becomes easier.

However, the importance of the standard form cannot be overstated. Without it, accurately graphing an ellipse from a mere equation can be challenging. The challenge we faced originally with \(x^2 + 4y^4 - 4 = 0\) highlights why being in the correct form can immediately communicate whether the equation can represent an ellipse or belongs to another group of conic sections. Understanding these differences is key to mastering conic sections.