To find the correct polar equation of the conic given the eccentricity [tex]\( e = 1.6 \)[/tex] and the directrix [tex]\( y = 4 \)[/tex], follow these steps:
1. Understand the Polar Equation of a Conic:
The standard forms of the polar equation for a conic with a directrix parallel to the coordinate axes are:
[tex]\[
r = \frac{ed}{1 + e \cos \theta} \quad \text{or} \quad r = \frac{ed}{1 + e \sin \theta}
\][/tex]
where [tex]\( e \)[/tex] is the eccentricity and [tex]\( d \)[/tex] is the distance from the pole to the directrix.
2. Identify the Directrix and the Relevant Trigonometric Function:
Since the directrix is given as [tex]\( y = 4 \)[/tex], it is a horizontal line. For horizontal directrices:
- Use the sine function in the denominator because the directrix is parallel to the [tex]\( x \)[/tex]-axis:
[tex]\[
r = \frac{ed}{1 + e \sin \theta}
\][/tex]
3. Calculate the Numerator [tex]\( ed \)[/tex]:
Substitute the given values [tex]\( e = 1.6 \)[/tex] and [tex]\( d = 4 \)[/tex]:
[tex]\[
ed = 1.6 \times 4 = 6.4
\][/tex]
4. Form the Polar Equation:
Substitute the values into the polar equation format:
[tex]\[
r = \frac{6.4}{1 + 1.6 \sin \theta}
\][/tex]
5. Verify Among the Given Options:
Among the given options, the correct polar equation is:
[tex]\[
r = \frac{6.4}{1 + 1.6 \sin \theta}
\][/tex]
So, the correct answer is:
[tex]\[
r = \frac{6.4}{1 + 1.6 \sin \theta}
\][/tex]
