Determine the equation of the directrix of [tex]$r=\frac{26.4}{4+4.4 \cos \theta}$[/tex].
Answer (2)
Rewrite the polar equation in standard form by dividing the numerator and denominator by 4: r = 1 + 1.1 c o s θ 6.6 .
Identify the eccentricity e = 1.1 and e p = 6.6 .
Solve for p : p = 1.1 6.6 = 6 .
Determine the equation of the directrix: x = 6 .
Explanation
Problem Analysis We are given the polar equation of a conic section: r = 4 + 4.4 c o s θ 26.4 . Our goal is to find the equation of the directrix.
Rewrite in Standard Form First, we need to rewrite the given polar equation in the standard form: r = 1 + e c o s θ e p , where e is the eccentricity and p is the distance from the focus (pole) to the directrix. To do this, we divide both the numerator and the denominator of the given equation by 4:
Perform the Division Dividing by 4, we get: r = ( 4 + 4.4 c o s θ ) /4 26.4/4 = 1 + 1.1 c o s θ 6.6
Identify Eccentricity and ep Now, we can identify the eccentricity e and the product e p from the standard form. Comparing r = 1 + 1.1 c o s θ 6.6 with r = 1 + e c o s θ e p , we have: e = 1.1 and e p = 6.6
Solve for p Next, we solve for p using the values of e and e p :
p = e e p = 1.1 6.6 = 6
Determine the Directrix Equation Since the equation has the form r = 1 + e c o s θ e p , the directrix is a vertical line to the right of the pole (focus) at x = p . Therefore, the equation of the directrix is x = 6 .
Final Answer The equation of the directrix is x = 6 .
Examples
Understanding conic sections and their directrices is crucial in various fields, such as astronomy, where the orbits of planets and comets can be described using conic sections with the sun at one focus. The directrix helps define the shape and orientation of these orbits. For example, knowing the directrix of a satellite's elliptical orbit helps engineers predict its path and ensure it stays within the desired parameters, preventing it from drifting off course or colliding with other objects.
The equation of the directrix for the polar equation r = 4 + 4.4 c o s θ 26.4 is x = 6 . This is determined by rewriting the equation in standard form and identifying the relevant parameters for eccentricity and directrix distance. Ultimately, we find p = 6 , which defines the location of the directrix line.
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