Problem 51 - A Graphical Approach to College Algebra

Question

Write an equation for each conic. Each parabola has vertex at the origin, and each ellipse or hyperbola is centered at the origin. $$\text { Focus }(3,0) ; e=\frac{1}{2}$$

Step-by-Step Solution

Answer

The equation of the ellipse is \( \frac{x^2}{36} + \frac{y^2}{27} = 1 \).

Step 1: Determine the Conic Type with Given Data

The conic with the given information (a focus and eccentricity) can either be an ellipse or a hyperbola. Since the eccentricity \( e = \frac{1}{2} \) is less than 1, it is an ellipse.

Step 2: Use the Definition of Eccentricity for an Ellipse

The eccentricity \( e \) of an ellipse is given by \( e = \frac{c}{a} \), where \( c \) is the distance from the center to a focus, and \( a \) is the semimajor axis. We know \( c = 3 \) from the focus \((3,0)\), and \( e = \frac{1}{2} \). Solve for \( a \): \[ \frac{3}{a} = \frac{1}{2} \] Clearing the fraction gives: \[ 3 = \frac{a}{2} \] Multiplying both sides by 2, we get \( a = 6 \).

Step 3: Find the Length of the Minor Axis

For ellipses, the relationship between the axes is \( a^2 = b^2 + c^2 \). From Step 2, we know \( a = 6 \) and \( c = 3 \). Substituting into the equation gives: \[ 6^2 = b^2 + 3^2 \] \[ 36 = b^2 + 9 \] Solving for \( b^2 \), subtract 9 from both sides: \[ b^2 = 27 \] Thus, \( b = \sqrt{27} = 3\sqrt{3} \).

Step 4: Write the Equation of the Ellipse

The standard form equation of an ellipse centered at the origin with the major axis along the x-axis is: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] From previous steps, we have \( a = 6 \) and \( b^2 = 27 \). Substituting these values, the equation becomes: \[ \frac{x^2}{36} + \frac{y^2}{27} = 1 \]

Key Concepts

Ellipse Equation

An ellipse is a type of conic section that can be understood as a flattened circle. To find the equation of an ellipse with its center at the origin, we use the standard form: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] Here, \(a\) represents the semi-major axis, and \(b\) is the semi-minor axis. These are essentially the "radii" of the ellipse in the horizontal and vertical directions, respectively. If an ellipse is aligned along the x-axis, then \(a^2\) corresponds to the denominator of the \(x^2\) term and \(b^2\) to the \(y^2\) term, and vice versa if it is aligned along the y-axis.

• Given a focus (3,0) and eccentricity \(e = \frac{1}{2}\), the ellipse has \(c = 3\).
• With \(e = \frac{c}{a}\), solving for \(a\) confirms it as 6.
• The relationship \(a^2 = b^2 + c^2\) helps find \(b^2 = 27\), leading to the ellipse equation: \(\frac{x^2}{36} + \frac{y^2}{27} = 1 \).

Eccentricity

Eccentricity is a measure that helps us understand how much a conic section deviates from being a circle. For different conics, eccentricities vary:

• Circles have an eccentricity of zero.
• Ellipses have eccentricities between 0 and 1.
• Parabolas have an eccentricity of exactly 1.
• Hyperbolas have eccentricities greater than 1.

For an ellipse, the eccentricity is calculated as \(e = \frac{c}{a}\), where:

• \(c\) is the distance from the center to any of the foci.
• \(a\) is the length of the semi-major axis.
• An eccentricity of \(\frac{1}{2}\) indicates the ellipse is not too stretched, maintaining a certain roundness.

Foci of Conics

The foci (plural for focus) of a conic section are crucial in defining its shape and other properties. In an ellipse, which resembles an elongated circle:

• The foci are two fixed points located symmetrically along the major axis.
• The sum of the distances from any point on the ellipse to the two foci remains constant.
• This unique property of ellipses ensures their shape, derived using the constant: \(2a\).

For the given problem, where one focus is \((3,0)\), these foci guide how the ellipse stretches along its main direction:

• If the major axis is horizontal, the foci lie along the x-axis.
• If the major axis is vertical, they are positioned along the y-axis.
• The distance from the center to one focus is \(c = 3\), critical for calculating the ellipse equation.

Understanding foci not only enhances your grasp of the basic properties of ellipses but also aids in solving related geometric problems.