Problem 67 - Precalculus

Question

Determine whether the statement is true or false. Justify your answer. The conic represented by the following equation is an ellipse. $$r^{2}=\frac{16}{9-4 \cos \left(\theta+\frac{\pi}{4}\right)}$$

Step-by-Step Solution

Answer

False. The conic represented by the equation is not an ellipse, as its eccentricity is greater than 1.

Step 1: Identify Parameters

Rewrite the equation in the standard form for a conic in polar coordinates \(r^2=\frac{a^2}{1\pm e\cos(\theta-\phi)}\). Look for the semi-major axis a as the denominator's constant (16 in this case) and the constant accompanying the cosine function in the denominator (4 in this case, but the absolute value is used for eccentricity, so eccentricity e is 2). The polar angle \(\phi\) is also found in the cosine function; in this case, it's \(-\pi/4\).

Step 2: Examine Eccentricity

Looking at the eccentricity, remember that if e < 1, this is an ellipse. The calculated eccentricity is 2 which is greater than 1 hence doesn't satisfy the condition of an ellipse.

Step 3: Final Conclusion

Since the eccentricity value found does not support the condition for a conic section to be an ellipse (which is e < 1), then the statement 'The conic represented by the equation is an ellipse' is false.

Key Concepts

Conic Sections

Conic sections are curves formed by the intersection of a plane with a double napped cone. These shapes play a crucial role in mathematics and include ellipses, parabolas, hyperbolas, and circles. Each conic section has unique properties and can be represented in different forms, depending on the coordinate system used to describe them.
In polar coordinates, conic sections are expressed using the formula:\[r = \frac{e \, a}{1 \pm e \cos(\theta - \phi)},\]where:

• \(r\) is the radius,
• \(e\) is the eccentricity,
• \(a\) is a constant known as the semi-major axis,
• and \(\phi\) is the angle of the major axis with respect to the polar coordinate system.

Understanding these components is essential to identify whether a given polar equation represents a specific type of conic section.

Ellipse

An ellipse is one of the four fundamental types of conic sections. It is an elongated circle with unique geometric properties. In the context of polar coordinates, an ellipse has a specific condition,where its eccentricity \(e\) must be less than 1.

• The formula for an ellipse in polar coordinates is \( r = \frac{a (1 - e^2)}{1 - e \cos(\theta)}\),
• where the semi-major axis \(a\) represents the longest radius of the ellipse, and
• the semi-minor axis is smaller than \(a\), indicating the ellipse's "flattened" shape.

Ellipses have two fixed points known as foci, and any point on the ellipse maintains a constant sum of distances to these foci. This distinctive property makes ellipses prominent in physics, especially in the orbits of planets.

Eccentricity

Eccentricity is a critical parameter in defining the shape of conic sections. This value indicates how "stretched" a conic section is compared to a perfect circle.

• For a circle, the eccentricity \(e = 0\),
• for an ellipse, \(0 < e < 1\),
• a parabola has an eccentricity equal to 1, and
• hyperbolas have an eccentricity greater than 1.

The specific range of eccentricity dictates the type of conic section that will be formed. For example, an eccentricity of 2 as seen in the exercise suggests a hyperbola, not an ellipse. This misunderstanding often leads to incorrect identification of conic sections, so being precise with eccentricity values is crucial for accurate descriptions.