Express the following polar equation of a conic to rectangular general form.−22−2cosθ

1. Break Down the Problem

We need to convert the polar equation given in the form: $r = -\frac{22}{1 - 2 \cos \theta}$ to rectangular general form $Ax + By + C = 0$.

2. Relevant Concepts

We will use the relationships between polar and rectangular coordinates:

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

3. Analysis and Detail

1. Start with the polar equation: $r = -\frac{22}{1 - 2 \cos \theta}$

2. Multiply both sides by $1 - 2 \cos \theta$: $r(1 - 2 \cos \theta) = -22$

3. Expanding the left-hand side gives: $r - 2r \cos \theta = -22$

4. Substitute $r$ with $\sqrt{x^2 + y^2}$ and $\cos \theta$ with $\frac{x}{r}$: $r - 2\left(\sqrt{x^2 + y^2}\right) \left(\frac{x}{\sqrt{x^2 + y^2}}\right) = -22$

5. This simplifies to: $r - 2x = -22$ Replace $r$ with $\sqrt{x^2 + y^2}$: $\sqrt{x^2 + y^2} - 2x = -22$

6. Rearranging gives: $\sqrt{x^2 + y^2} = 2x - 22$

7. Squaring both sides to eliminate the square root: $x^2 + y^2 = (2x - 22)^2$

8. Expanding the right-hand side: $x^2 + y^2 = 4x^2 - 88x + 484$

9. Rearranging the equation: $x^2 + y^2 - 4x^2 + 88x - 484 = 0$ which simplifies to: $-3x^2 + y^2 + 88x - 484 = 0$

4. Verify and Summarize

We have transformed the original polar equation into rectangular form. It simplifies to: $3x^2 - y^2 - 88x + 484 = 0$

Final Answer

The rectangular general form of the given polar equation is: $3x^2 - y^2 - 88x + 484 = 0$