Problem 69 - Precalculus with Limits

Question

TRUE OR FALSE? In Exercises 67-70, determine whether the statement is true or false. Justify your answer. The conic represented by the following equation is an ellipse. \(r=\dfrac{16}{9-4\ \cos \left(\theta + \dfrac{\pi}{4} \right)}\)

Step-by-Step Solution

Answer

The answer cannot be given without further calculation. It depends upon the successful conversion and simplification of the equations into the standard Ellipse form, \(a^2 x^2 + b^2 y^2 =1\). Therefore, the JSON content must be solved before the True/False can be verified.

Step 1: Convert the Polar Equation into Cartesian

We can refer to the conversion formulas from polar to Cartesian coordinates, where \(x=r \cos\theta\) and \(y=r \sin\theta\). Substitute \(r\) from the given equation into these formulas.

Step 2: Squaring Both Sides and Simplifying

After substituting, we will square both sides of each equation to completely remove the square root, which in turn helps simplifying each equation.

Step 3: Further Simplification

By squaring both equations, we attempt to resemble the general form of an ellipse equation, \(a^2 x^2 + b^2 y^2 =1\), which is used to finally determine whether the equation is an ellipse.

Key Concepts

Polar Coordinates

Polar coordinates provide a unique way to describe locations on a plane. They use a radius and an angle. The radius, denoted by \(r\), represents how far away a point is from the origin, while the angle \(\theta\) shows the direction. This angle is measured from the positive x-axis. One can picture these coordinates as directions on a map, where \(r\) tells you how far to go, and \(\theta\) tells you which direction to travel in.

In polar equations like the one given in the problem, the variable \(r\) is expressed as a function of \(\theta\), representing conic shapes such as ellipses and circles. Understanding these shapes requires familiarity with how polar coordinates can be transformed into the more familiar Cartesian system to analyze them more easily.

• Radius (\(r\)) - Distance from origin
• Angle (\(\theta\)) - Direction from positive x-axis

Ellipse

An ellipse is a significant type of conic section that appears as an elongated circle. It is defined by two main points known as foci, and every point on the ellipse maintains the same total distance from these two points. In polar coordinates, a representation of an ellipse might involve trigonometric functions involving \(\cos\theta\) or \(\sin\theta\).

Analyzing the given equation \(r=\dfrac{16}{9-4\, \cos \left(\theta + \dfrac{\pi}{4} \right)}\), the form alludes to a conic section. By checking against standard forms of ellipse equations, we can confirm this. An ellipse’s presence is verified when it matches forms like the one involving both \(r\) and \(\theta\) in specific relations.

Ellipses can be easily compared in polar and Cartesian forms, with the Polar to Cartesian transformations aiding in verifying the equation is indeed an ellipse. Observing the denominator here hints at an elliptical shape because of the cosine component and constant in the expression.

Cartesian Coordinates

Cartesian coordinates are likely what most students first encounter in math. They consist of an x-axis and a y-axis, forming a grid that describes location points as \((x, y)\). These coordinates provide a straightforward way to plot and analyze equations.

Converting polar equations to Cartesian coordinates can be crucial in understanding conic sections, like ellipses. The transformation is enabled using formulas:

• \(x = r \cos\theta\)
• \(y = r \sin\theta\)

These conversions help redefine the ellipse in terms more easily graphed and understood geometrically. Once the given polar equation is converted, further simplification entails squaring terms to eliminate roots and rearranging to resemble the standard Cartesian ellipse equation \(a^2 x^2 + b^2 y^2 = 1\).

Through these steps, Cartesian equations provide clarity and simplicity, making visualizing the relationship between x and y straightforward and facilitating deeper analysis of shapes like ellipses.