Problem 48 - College Algebra

Question

For the following exercises, use the given information about the graph of each ellipse to determine its equation. Center at the origin, symmetric with respect to the \(x\) - and \(y\) -axes, focus at \((3,0),\) and major axis is twice as long as minor axis.

Step-by-Step Solution

Answer

The equation of the ellipse is \( \frac{x^2}{12} + \frac{y^2}{3} = 1 \).

Step 1: Identify Ellipse Characteristics

The given ellipse is centered at the origin \(0, 0\). The ellipse is symmetric with respect to both the \(x\) and \(y\)-axes. This symmetry means it is aligned with the coordinate axes, so it is either horizontal or vertical. Since the focus is at \(3, 0\), the major axis is horizontal.

Step 2: Determine Major and Minor Axes

We know the major axis is along the \(x\)-axis (horizontal) because the focus is \(3, 0\). The major axis length is determined to be twice that of the minor axis. If the length of the minor axis is \(2b\), then the major axis is \(4b\).

Step 3: Recognize Ellipse Formula

For a horizontally oriented ellipse centered at the origin, the standard form of the equation is \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]where \(a\) is the semi-major axis and \(b\) is the semi-minor axis with \(a > b\).

Step 4: Calculate Semi-Major and Semi-Minor Axes

Since the distance from the center to the focus (c) is \(3\), and using the relationship \(c^2 = a^2 - b^2\), we determine \(c = 3\), \(a = 2b\). Thus:\[ 3^2 = (2b)^2 - b^2 \]\[ 9 = 4b^2 - b^2 \]\[ 9 = 3b^2 \]\[ b = \frac{3}{\sqrt{3}} = \sqrt{3} \]

Step 5: Confirm Values of 'a' and 'b' and Write Equation

Substitute \(b = \sqrt{3}\) back to find \(a\): \[ a = 2b = 2\sqrt{3} \]Place the values of \(a\) and \(b\) in the ellipse equation:\[ \frac{x^2}{(2\sqrt{3})^2} + \frac{y^2}{(\sqrt{3})^2} = 1 \]\[ \frac{x^2}{12} + \frac{y^2}{3} = 1 \]

Key Concepts

Conic Sections

Conic sections, often referred to as "conics," are the curves obtained by slicing a double-napped cone with a plane. This intriguing family of curves includes ellipses, parabolas, hyperbolas, and circles. Each type is determined by the angle and position of the slice relative to the cone's axis.
Ellipses come into play when the intersecting plane cuts through the cone at an angle, not parallel to the base, and does not pass through the apex. They appear as elongated circles with two focal points.
Understanding how these conic sections arise helps in visualizing their properties. In our current exercise, we're dealing with an ellipse—a key member of the conic sections family.

Ellipse Characteristics

Ellipses have several defining characteristics that set them apart from other conic sections. Key attributes include:

• Symmetry: An ellipse is symmetric about its major and minor axes.
• Center: The middle point from which the distances to each focus are equal.
• Foci: Two points inside the ellipse, equidistant from the center, guiding its shape.
• Axes: The longest diameter (major axis) and the shortest diameter (minor axis) define the shape and size.

In our example, the ellipse is centered at the origin and symmetric about the x- and y-axes. With the focus at (3, 0) and a major axis that's twice the length of the minor axis, this determines its layout precisely. These characteristics all play crucial roles in establishing the ellipse's equation.

Standard Form of Ellipse

The standard form of an ellipse's equation is tailored to its orientation and dimensions. For an ellipse centered at the origin, it appears as:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
where \(a\) and \(b\) are the semi-major and semi-minor axes respectively. For horizontally aligned ellipses, \(a > b\), emphasizing the major axis runs along the x-axis.
Calculating \(a\) and \(b\) involves understanding their relationship with the ellipse's foci. For our ellipse, the focus at (3, 0) indicates a semi-major axis of 2\(b\) and a focus length \(c\) of 3. Through the equation \(c^2 = a^2 - b^2\), values for \(a\) and \(b\) were derived as \(2\sqrt{3}\) and \(\sqrt{3}\) respectively.
Plugging these into the standard form gives the final equation \[ \frac{x^2}{12} + \frac{y^2}{3} = 1 \]. This representation concisely captures the ellipse's unique properties.