Problem 15 - Contemporary Precalculus

Question

Identify the conic section whose equation is given and find its graph. If it is a circle, list its center and radius. If it is an ellipse, list its center, vertices, and foci. $$4 x^{2}+3 y^{2}=12$$

Step-by-Step Solution

Answer

Question: Identify the conic section represented by the equation $$4x^2 + 3y^2 = 12$$ and find its center, vertices, and foci. Answer: The given equation represents an ellipse with a center at (0, 0), vertices at (±2, 0), and foci at (±1, 0).

Step 1: Rewrite the equation

Divide each term by 12 to write the given equation in the standard form of an ellipse equation: $$\frac{4 x^{2}}{12}+\frac{3 y^{2}}{12}=1$$

Step 2: Simplify the equation

Simplify each fraction to find the standard form of the ellipse equation: $$\frac{x^{2}}{3}+\frac{y^{2}}{4}=1$$

Step 3: Identify the center, a, and b

From the standard form of the ellipse equation, we can identify the center \((h, k) = (0, 0)\), and the semi-major axis, \(a =\sqrt{4} = 2\), and the semi-minor axis, \(b =\sqrt{3}\).

Step 4: Find the vertices

Use the semi-major axis (a) and center to find the vertices of the ellipse: Vertices: \((±a, 0) = (±2, 0)\)

Step 5: Find the foci

Use the relationship \(c = \sqrt{a^2 - b^2}\) to find the distance between the center and each focus: \(c = \sqrt{2^2 - \sqrt{3}^2}\) \(c = \sqrt{4 - 3}\) \(c = \sqrt{1}\) \(c = 1\) Foci: \((±c, 0) = (±1, 0)\) The ellipse has a center at \((0, 0)\), vertices at \((±2, 0)\), and foci at \((±1, 0)\).

Key Concepts

Ellipse Equation

An ellipse is a fascinating conic section that is often written in equation form to understand its shape and properties. The standard equation of an ellipse that is centered at the origin \(0,0\) can be given by \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively. The important thing to remember is that this equation equates to 1, after simplifying and adjusting the coefficients of \(x^2\) and \(y^2\).

• If \(a > b\), the ellipse is longer along the x-axis.
• If \(b > a\), it's longer along the y-axis, known as a vertical ellipse.

A key step in understanding any ellipse is transforming its equation into this standard form.

Standard Form

The standard form of an ellipse helps quickly identify its crucial features. To achieve this, you must manipulate the given equation to fit the format \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). For instance, take the equation \( 4x^2 + 3y^2 = 12 \). By dividing each term by 12, we get \( \frac{4x^2}{12} + \frac{3y^2}{12} = 1 \).
Simplifying this gives us \( \frac{x^2}{3} + \frac{y^2}{4} = 1 \).
Now, you can identify \(a = \sqrt{4}\) and \(b = \sqrt{3}\).
It's essential to simplify the fractions correctly so you can accurately determine \(a\) and \(b\).

Center of an Ellipse

The center of an ellipse plays an essential role in defining its structure and position in the coordinate system. In the standard form expressed as \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), the center is identified by the coordinates \(h\) and \(k\).
For our example, \( \frac{x^2}{3} + \frac{y^2}{4} = 1 \), the center \(h, k\) is \(0, 0\).
This means the ellipse is centered at the origin, which simplifies determining the vertices and foci because \(x\) and \(y\) coordinates in those features are relative to \((h, k)\).

• The center acts as the midpoint from where distances to vertices and foci are measured.

Vertices of an Ellipse

Vertices are the points on an ellipse that lie along its major axis at the maximum distance from the center. For an ellipse described by \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), the vertices can be found using the semi-major axis.
In our case, with \(a = 2\), vertices are at \( (\pm2, 0) \).
Since \(a > b\) - this tells us the major axis is along the x-axis. Thus, the vertices are horizontally aligned, equidistant around the center.

• Vertices are found symmetrically on both sides of the center.
• They define the length of the major axis, which is \(2a\).

Identifying vertices is a crucial step in sketching the ellipse and understanding its orientation.

Foci of an Ellipse

The foci (singular: focus) of an ellipse have a unique role in its geometric definition. An ellipse can be thought of as the set of all points for which the sum of the distances to two fixed points (the foci) is constant.
For an ellipse in the form of \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), the distance \(c\) from the center to each focus is calculated using \( c = \sqrt{a^2 - b^2} \).
In our case, \(c = \sqrt{2^2 - \sqrt{3}^2} = 1\).
Thus, the foci are placed at \( (\pm1, 0) \).

• The foci give us more than just points to plot; they are essential in understanding the geometry of ellipses.
• The closer the foci are to the center, the more circle-like, or less elongated, the ellipse is.

Understanding these helps grasp the shape and behavior of the ellipse.