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Chapter 10: Problem 36

If you are given the standard form of the polar equation of a conic, how do you determine the location of a directrix from the focus at the pole?

Short Answer

Expert verified
To find the location of the directrix from the focus at the pole in a conic's standard polar equation, one must identify the eccentricity \(e\) and the distance from the focus to the directrix \(p\), as well as set the appropriate angle \( \theta \). The location can then be calculated with the equation \( \dfrac{ep}{1+\cos(\theta)} \).

Step by step solution

01

Identifying the eccentricity and distance

Identify the values of the eccentricity \(e\) and the distance from the focus to the directrix \(p\). These values are given by the standard form of the polar equation of the given conic.
02

Setting the appropriate angle

Set the value of the angle \( \theta \). In general, to obtain the equation of the directrix, we don't need a specific value for the angle. However, if the location of a specific directrix is requested, then the appropriate angle must be used. This angle is the one formed by the x-axis and the line connecting the focus and any point on the conic.
03

Use the equation

Use the equation \( \dfrac{ep}{1+\cos(\theta)} \) to find the location of the directrix. This equation allows us to find the specific location of the directrix relative to the focus that is at the pole.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. You told me that an ellipse centered at the origin has vertices at \((-5,0)\) and \((5,0),\) so 1 was able to graph the ellipse.

Describe a viewing rectangle, or window, such as [-30, 30, 3] by [-8, 4, 1], that shows a complete graph of each polar equation and minimizes unused portions of the screen. $$ r=\frac{16}{3+5 \cos \theta} $$

a. Make a sketch of an angle \(\theta\) in standard position for which $$ \cot 2 \theta=-\frac{7}{24} \text { and } 90^{\circ}<2 \theta<180^{\circ} $$ b. Use your sketch from part (a) to determine the value of cos \(2 \theta\) c. Use the value of \(\cos 2 \theta\) from part (b) and the identities $$ \sin \theta=\sqrt{\frac{1-\cos 2 \theta}{2}} \text { and } \cos \theta=\sqrt{\frac{1+\cos 2 \theta}{2}} $$ to determine the values of \(\sin \theta\) and \(\cos \theta\) d. In part (c), why did we not write \(\pm\) before the radical in each formula?

In Exercises \(51-60,\) convert each equation to standard form by completing the square on \(x\) and \(y .\) Then graph the ellipse and give the location of its foci. $$ 4 x^{2}+25 y^{2}-24 x+100 y+36=0 $$

In Exercises \(67-68,\) graph each semiellipse. $$ y=-\sqrt{16-4 x^{2}} $$
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