Explore a world of knowledge and get your questions answered on IDNLearn.com. Receive prompt and accurate responses to your questions from our community of knowledgeable professionals ready to assist you at any time.

Write a polar equation for the conic with the given characteristics.

Given:
[tex]\[ e = 1.6 \][/tex]
Directrix: [tex]\[ y = 4 \][/tex]

A. [tex]\( r = \frac{6.4}{1 + 1.6 \cos \theta} \)[/tex]
B. [tex]\( r = \frac{6.4}{1 + 1.6 \sin \theta} \)[/tex]
C. [tex]\( r = \frac{4}{1 + 1.6 \sin \theta} \)[/tex]
D. [tex]\( r = \frac{4}{1 + 1.6 \cos \theta} \)[/tex]


Sagot :

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]