To determine the equation of the directrix for the given polar equation of a conic section [tex]\( r = \frac{26.4}{4 + 4.4 \cos \theta} \)[/tex], follow these steps:
1. Identify the Conic Section and Parameters:
The given equation is of the form [tex]\( r = \frac{ed}{1 + e \cos \theta} \)[/tex], where [tex]\( e \)[/tex] is the eccentricity and [tex]\( d \)[/tex] is the distance from the focus to the directrix.
2. Relate the Given Equation to the Standard Form:
By comparing [tex]\( r = \frac{26.4}{4 + 4.4 \cos \theta} \)[/tex] with [tex]\( r = \frac{ed}{1 + e \cos \theta} \)[/tex], we can see that:
[tex]\[
26.4 = ed \quad \text{and} \quad 4 + 4.4 \cos \theta = 1 + e \cos \theta.
\][/tex]
3. Solve for Eccentricity [tex]\( e \)[/tex]:
From the equation [tex]\( 4 + 4.4 \cos \theta = 1 + e \cos \theta \)[/tex], we can infer:
[tex]\[
e \cos \theta = 4.4 \cos \theta + 3.
\][/tex]
By comparing coefficients of [tex]\(\cos \theta\)[/tex] on both sides, we get:
[tex]\[
e = 4.4.
\][/tex]
4. Determine the Distance [tex]\( d \)[/tex]:
Using the eccentricity [tex]\( e = 4.4 \)[/tex] and the relationship [tex]\( 26.4 = ed \)[/tex]:
[tex]\[
26.4 = 4.4d.
\][/tex]
Solving for [tex]\( d \)[/tex]:
[tex]\[
d = \frac{26.4}{4.4} = 6.
\][/tex]
5. Equation of the Directrix:
For a conic section with the focus at the origin and using the eccentricity [tex]\( e \)[/tex] and distance [tex]\( d \)[/tex]:
- When [tex]\( e \cos \theta = 1 + e = 4.4 + 1 = 5.4 \)[/tex].
- The directrix is vertical:
- When [tex]\( e = 4.4 \)[/tex], cos \theta = 1 ( [tex]\(\theta = 0 \)[/tex] ).
- [tex]\( x = -d \)[/tex] for the directrix.
Thus, the equation for the directrix considering the [tex]\(d = 10.171428571428569\)[/tex] is:
[tex]\[
\boxed{x = -6}.
\][/tex]
