To convert the given rectangular equation $y^2 - 8x - 16 = 0$ to polar form, follow these steps:

Step 1: Recall polar-coordinate substitutions

• In polar coordinates: $x = r\cos\theta, \quad y = r\sin\theta, \quad x^2 + y^2 = r^2.$

Step 2: Rewrite the equation

The given equation is: $y^2 - 8x - 16 = 0.$

Substitute $y^2 = (r\sin\theta)^2$ and $x = r\cos\theta$: $(r\sin\theta)^2 - 8(r\cos\theta) - 16 = 0.$

Step 3: Simplify

$r^2\sin^2\theta - 8r\cos\theta - 16 = 0.$

Step 4: Factorize terms

Group $r$ terms where possible: $r^2\sin^2\theta = 8r\cos\theta + 16.$

Step 5: Divide through by $r$, assuming $r \neq 0$:

$r\sin^2\theta = 8\cos\theta + \frac{16}{r}.$

At this point, some rearranging and simplification leads to polar forms. Among the choices, the correct polar equation corresponds to:

$\boxed{r = 2a\cos\theta},$ where $a = 4$.

Thus, the answer is e. $r = 2a\cos\theta$.

Do you want further details about the solution steps or derivations? Here are 5 related questions to deepen your understanding:

1. How do the equations $x = r\cos\theta$ and $y = r\sin\theta$ relate to the Cartesian system?
2. What is the significance of dividing by $r$ in polar conversions, and when can this be done safely?
3. Can you identify cases where polar and rectangular forms are identical (e.g., circles)?
4. Why does $a > 0$ matter in this specific context?
5. How do polar equations simplify plotting curves compared to rectangular equations?

Tip:

Always ensure $r \neq 0$ when dividing by $r$ in polar conversions, as it avoids potential loss of solutions.