To convert the given polar equation $r = 4 \sin \theta - 6 \cos \theta$ into rectangular form, we can use the following standard polar-to-rectangular conversions:

• $r = \sqrt{x^2 + y^2}$
• $x = r \cos \theta$
• $y = r \sin \theta$

We will also need the identity relations for the trigonometric functions:

• $\sin \theta = \frac{y}{r}$
• $\cos \theta = \frac{x}{r}$

Step-by-Step Conversion:

1. Start with the given equation: $r = 4 \sin \theta - 6 \cos \theta$

2. Replace $\sin \theta$ and $\cos \theta$ using their rectangular equivalents: $r = 4 \left(\frac{y}{r}\right) - 6 \left(\frac{x}{r}\right)$

3. Multiply both sides by $r$ to eliminate the denominator: $r^2 = 4y - 6x$

4. Now, recall that $r^2 = x^2 + y^2$, so we substitute this into the equation: $x^2 + y^2 = 4y - 6x$

5. Rearrange the equation to set everything to one side: $x^2 + y^2 + 6x - 4y = 0$

6. Complete the square for both the $x$-terms and $y$-terms:

   • For $x$, take half of 6 and square it: $\left(\frac{6}{2}\right)^2 = 9$, so we add and subtract 9.
   • For $y$, take half of 4 and square it: $\left(\frac{4}{2}\right)^2 = 4$, so we add and subtract 4.

   The equation becomes: $(x^2 + 6x + 9) + (y^2 - 4y + 4) = 9 + 4$ Simplifying: $(x + 3)^2 + (y - 2)^2 = 13$

This is the equation of a circle with center $(-3, 2)$ and radius $\sqrt{13}$.

Conclusion:

The correct rectangular form is: $(x + 3)^2 + (y - 2)^2 = 13$

Thus, the correct answer is C.

Would you like further details or have any questions?

Here are 5 related questions for deeper understanding:

1. How do you convert between polar and rectangular coordinates?
2. What is the geometric interpretation of the equation $r = 4 \sin \theta - 6 \cos \theta$?
3. How does completing the square help in solving conic sections?
4. Can you generalize how to find the center and radius of a circle from its equation?
5. What other types of equations can be converted between polar and rectangular forms?

Tip: Remember that completing the square is a useful technique when transforming quadratic equations into standard forms like circles or parabolas.