Questions: Convert the given polar equation to a Cartesian equation. (Use the following as necessary: [ r=frac4sin (theta)+5 cos (theta) ]

Solution Steps

To convert the given polar equation to a Cartesian equation, we need to use the relationships between polar and Cartesian coordinates: \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \). We can express \( \sin(\theta) \) and \( \cos(\theta) \) in terms of \( x \) and \( y \) using these relationships, and then substitute them into the given polar equation to express it in terms of \( x \) and \( y \).

The given polar equation is

\[ r = \frac{4}{\sin(\theta) + 5 \cos(\theta)} \]

Using the relationships between polar and Cartesian coordinates, we have:

\[ \sin(\theta) = \frac{y}{r} \quad \text{and} \quad \cos(\theta) = \frac{x}{r} \]

Substituting these expressions into the polar equation gives:

\[ r = \frac{4}{\frac{y}{r} + 5 \cdot \frac{x}{r}} \]

Multiplying both sides by \(r\) to eliminate the denominator results in:

\[ r^2 = 4 \cdot \frac{1}{\frac{y}{r} + 5 \cdot \frac{x}{r}} \cdot r \]

This simplifies to:

\[ r^2 = 4 \cdot \frac{r}{y + 5x} \]

Rearranging gives:

\[ r(y + 5x) = 4 \]

Since \(r = \sqrt{x^2 + y^2}\), we substitute this into the equation:

\[ \sqrt{x^2 + y^2}(y + 5x) = 4 \]

Final Answer