Question

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

Short 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-by-step solution

Write the given ellipse equation

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

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.

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.

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 \(y'\).