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Chapter 6: Problem 20

A point in rectangular coordinates is given. Convert the point to polar coordinates. $$(0,-5)$$

Short Answer

Expert verified
The polar coordinates of the point (0, -5) in rectangular coordinates are (5, -蟺/2) or (5, -90掳)

Step by step solution

01

Identify the rectangular coordinates.

The rectangular coordinates are given as (0, -5). This means that the x-coordinate is 0 and the y-coordinate is -5.
02

Calculate the radius.

The radius in polar coordinates is calculated from rectangular coordinates using the formula \(r = \sqrt{ x^{2} + y^{2} }\). Substituting x=0 and y=-5 into the formula gives \(r = \sqrt{0 + (-5)^{2}} = \sqrt{25} = 5\). So, the radius is 5.
03

Calculate the angle.

The angle (theta 胃) in polar coordinates is calculated from rectangular coordinates. Since the y-coordinate is negative and x-coordinate is zero, the point lies on the negative y-axis, thus the angle 胃 is -90 degrees or -蟺/2 in radians.
04

Write the polar coordinates.

The polar coordinates are usually written in the form (r, 胃). Here, r=5 and 胃=-蟺/2 radians or -90 degrees. So, the polar coordinates are (5, -蟺/2) or (5, -90掳)

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Most popular questions from this chapter

\(r=1-2 \sin \theta\)

In Exercises 25-28, use a graphing utility to graph the polar equation. Identify the graph. $$ r=\frac{-5}{2+4 \sin \theta} $$

In Exercises 33-48, find a polar equation of the conic with its focus at the pole. $$ \begin{array}{lll} {\text { Conic }} & \text { Eccentricity } & \text { Directrix } \\ \text { Hyperbola } & e=\frac{3}{2} & x=-1 \end{array} $$

In Exercises 33-48, find a polar equation of the conic with its focus at the pole. $$ \begin{array}{ll} {\text { Conic }} & \text { Vertex or Vertices } \\ \text { Parabola } & (1,-\pi / 2) \\ \end{array} $$

Show that the polar equation of the hyperbola $$ \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \quad \text { is } \quad r^{2}=\frac{-b^{2}}{1-e^{2} \cos ^{2} \theta} . $$
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