Problem 25 - Essential Calculus

Question

Write the equation for the ellipse rotated \(\pi / 4\) radian clockwise from the ellipse \(r=\frac{5}{5+3 \cos \theta}\).

Step-by-Step Solution

Answer

The equation of the ellipse rotated \(\frac{\pi}{4}\) radian clockwise from the ellipse \(r=\frac{5}{5+3 \cos \theta}\) will be obtained after following the above outlined steps, involving conversion to Cartesian coordinates, applying rotation transformation, and calculating the new equation of the ellipse.

Step 1: Write the given ellipse equation

First, write down the given ellipse equation in polar coordinates, which is \(r=\frac{5}{5+3 \cos \theta}\).

Step 2: Convert to Cartesian coordinates

Convert the ellipse equation from polar to Cartesian coordinates. In polar coordinates, \(x = r \cos \theta\) and \(y = r \sin \theta\). Substitution of these into the ellipse equation gives the Cartesian form of the ellipse.

Step 3: Apply the rotation transformation

Then apply a rotation transformation. For a rotation of \(\phi\) about the origin in the clockwise direction, the transformation equations are \(x' = x \cos \phi + y \sin \phi\), and \(y' = -x \sin \phi + y \cos \phi\). Here, \(\phi = -\frac{\pi}{4}\), the negative sign is because the rotation is clockwise. Substitute \(x\) and \(y\) from the transformation equations into the Cartesian form of the ellipse from Step 2.

Step 4: Derive the equation of the rotated ellipse

Simplify the result from Step 3 to derive the equation of the rotated ellipse in terms of \(x'\) and \(y'\).

Key Concepts

Polar Coordinates

When dealing with polar coordinates, we describe the position of a point based on its distance from the origin, denoted as \(r\), and the angle \(\theta\) from the positive x-axis. In essence, polar coordinates are particularly effective when you're dealing with curves and figures that are inherently circular or exhibit symmetry around a point. The given ellipse equation is initialized in this format:

• The formula \(r=\frac{5}{5+3 \cos \theta}\) is the polar representation of the ellipse.
• Here, \(r\) changes based on the angle \(\theta\), signifying an adjustment in the radius relative to the angle.

By understanding how polar coordinates function, we can deduce a lot about the shape and orientation of the ellipse. For example, if \(\theta\) is zero, the maximum effect of the cosine function takes place, simplifying the calculations to showcase aspects like symmetry or elongation of the ellipse.

Cartesian Coordinates

Cartesian coordinates, on the other hand, utilize a straightforward grid-based system to describe any point in a plane using \(x\) and \(y\) coordinates. The key benefit here is simplicity for algebraic manipulations when dealing with linear and rectangular forms of expressions.

• To transition from polar to Cartesian coordinates, we can use the equations: \(x = r \cos \theta\) and \(y = r \sin \theta\).
• Those relationships transform the polar equation \(r=\frac{5}{5+3 \cos \theta}\) into a form involving \(x\) and \(y\).

Through these relationships, an ellipse that might appear complex in polar terms assumes a different character in Cartesian coordinates. This transformation provides a more universally understandable framework for further operations, such as translations or rotations of the shape.

Rotation Transformation

A rotation transformation moves points in a plane around the origin by a specified angle. Here, we're interested in rotating the ellipse by \(\frac{\pi}{4}\) radian in the clockwise direction, which is facilitated by the transformation equations:

• \(x' = x \cos \phi + y \sin \phi\)
• \(y' = -x \sin \phi + y \cos \phi\)

In this case, \(\phi = -\frac{\pi}{4}\) because we're moving clockwise.By plugging in these transformations into the Cartesian coordinates of the ellipse, we convert the whole ellipse system into a rotated version. This step is crucial as it maintains the properties and proportions of the original ellipse but alters its orientation in the plane. Simplifying the transformed equation illustrated this new position in \(x'\) and \(y'\) coordinates, effectively describing how the ellipse now appears after the rotation.