Problem 27 - Calculus Single Variable

Question

The center of an ellipse or hyperbola is the point of intersection of its axes of symmetry. Each state whether the graph of the given Cartesian equation is an ellipse or hyperbola. Determine its standard form and center. \(4 x^{2}+y^{2}=1\)

Step-by-Step Solution

Answer

The given equation represents an ellipse, with its standard form \(\frac{x^2}{1/4} + \frac{y^2}{1} = 1\) and center at \((0, 0)\).

Step 1: Identify the Given Equation Type

We have the equation \(4x^2 + y^2 = 1\). Both \(x\) and \(y\) are squared, and they have different coefficients, which suggests that this could either be an ellipse or a hyperbola. To differentiate, check if the signs associated with the squared terms are the same: they are both positive, so it is an ellipse.

Step 2: Rewrite in Standard Form

The general standard form of an ellipse is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). To achieve this, divide the entire given equation by 1 (since it is already set equal to 1): \(\frac{4x^2}{1} + \frac{y^2}{1} = 1\). Simplify by expressing each term with a denominator: \(\frac{x^2}{1/4} + \frac{y^2}{1} = 1\). Now our equation is in the form of \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) with \(a^2 = \frac{1}{4}\) and \(b^2 = 1\).

Step 3: Determine the Center of the Ellipse

An ellipse centered at the origin has the standard form \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\). Here, \(h\) and \(k\) represent the coordinates of the center. Our equation is already in this form without additional transformations or translations, indicating the center is at \((0, 0)\).

Key Concepts

Cartesian Equation of an Ellipse

A Cartesian equation is a mathematical expression that uses coordinates to describe geometric shapes. For ellipses, this equation helps us to see the curved shape by looking at the relationship between the squared terms of the variables involved. When faced with an equation like \(4x^2 + y^2 = 1\), our task is to understand whether this relationship shows an ellipse or another conic like a hyperbola.

When both \(x\) and \(y\) in the equation are squared and on the same side, we have hints pointing toward an ellipse. If the coefficients of these squared terms are positive, such as in our example, it further confirms this. A hyperbola, for instance, would have one positive and one negative squared term. So the simple check is:

• Are both squared terms positive? If yes, this suggests an ellipse.
• Check if they have different coefficients - this typically means an ellipse or a hyperbola could be present, waiting to be transformed into its standard shape for confirmation.

The moment you align these features, you've recognized the equation's type and can advance to find its more precise mathematical form.

Standard Form of an Ellipse Equation

Once we know we're dealing with an ellipse, the next step is rewriting the Cartesian equation into its standard form. The standard form allows us to extract further details about the ellipse's characteristics, such as its dimensions and orientation.

The general standard form of an ellipse is expressed as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). This structure helps us identify important numbers: \(a\) and \(b\), which directly relate to the ellipse's width and height, respectively.
In our example, we need to skillfully rewrite \(4x^2 + y^2 = 1\) by manipulating it to fit this framework:

• Divide through by 1; it stays unchanged as it is already equal to 1.
• Express \(\frac{4x^2}{1}\) as \(\frac{x^2}{1/4}\), thereby matching the standard form structure.
• Yields \(\frac{x^2}{1/4} + \frac{y^2}{1} = 1\), giving \(a^2 = \frac{1}{4}\) and \(b^2 = 1\).

These transformations and interpretations bring clarity to the mathematical presentation of the ellipse and enhance our understanding of its geometric profile.

Determining the Center of an Ellipse

In order to fully understand the ellipse's position, identifying its center is crucial. The center is defined as the point at which the ellipse is symmetrically balanced, serving as its origin in a transformed equation context.

The transformed mathematical standard form of an ellipse is noted by \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), where \((h, k)\) represents the center. When an ellipse is centered at the origin, as it is with our derived equation \(\frac{x^2}{1/4} + \frac{y^2}{1} = 1\), this simplifies to no translations:

• The values \(h = 0\) and \(k = 0\) show directly from the lack of transformations within the expression.
• This tells us that the center of the ellipse is located at the point \((0, 0)\).

By situating the center at \((0,0)\), we clearly understand the symmetry and formation location of the ellipse, emphasizing its balanced nature around this pivotal point.