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Chapter 9: Q. 64 (page 732)

Let $a 鈮� 0$ . Prove that the graph of the equation $r = \frac{a}{1 鈭� \cos 胃}$ is a parabola in a polar coordinate system. When $a 鈮� 0$ , what is the graph of the equation $r = \frac{a}{1 - \sin 胃}$ ?

Short Answer

Expert verified

It has been proved that graph of equation is a parabola.

The graph of $r = \frac{a}{1 - \sin 胃}$ is also a parabola.

Step by step solution

01

Step 1. Given information

$r = \frac{a}{1 - \cos 胃}$ , $a 鈮� 0$

02

Step 2. Convert polar form of equation into rectangular form.

$r = \frac{a}{1 - \cos 胃} r = \frac{a}{1 - \frac{x}{r}} r = \frac{a}{\frac{r - x}{r}} r = \frac{a r}{r - x} r - x = a r = x + a \sqrt{x^{2} + y^{2}} = x + a {(\sqrt{x^{2} + y^{2}})}^{2} = {(x + a)}^{2} x^{2} + y^{2} = x^{2} + a^{2} + 2 a x y^{2} = 2 a x + a^{2} y^{2} = 2 a (x + \frac{a}{2})$

This is the equation of parabola which is symmetrical about $x -$ axis.

Similarly graph of $r = \frac{a}{1 - \sin 胃}$ is also a parabola, which is symmetrical about $y -$ axis.

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Most popular questions from this chapter

Use polar coordinates to graph the conics in Exercises 44鈥�51.

$r = \frac{2}{1 - 3 \cos 胃}$

In Exercises 32鈥�47 convert the equations given in polar coordinates to rectangular coordinates.

$r^{2} = \sin 胃$

Use Cartesian coordinates to express the equations for the hyperbolas determined by the conditions specified in Exercises 38鈥�43.

$foci (3, 1) and (3, 9), directrix y = 4$

In Exercises 24鈥�31 find all polar coordinate representations for the point given in rectangular coordinates.

$(- 2, 0)$

Each of the integrals or integral expressions in Exercises $38 - 44$ represents the area of a region in the plane. Use polar coordinates to sketch the region and evaluate the expressions.

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