Converting a conic in polar form to rectangular form
Convert the conic
$\text{\hspace{0.17em}}r=\frac{1}{5-5\mathrm{sin}\text{\hspace{0.17em}}\theta}$ to rectangular form.
We will rearrange the formula to use the identities
$r=\sqrt{{x}^{2}+{y}^{2}},x=r\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta ,\text{and}y=r\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta .$
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Key concepts
Any conic may be determined by a single focus, the corresponding eccentricity, and the directrix. We can also define a conic in terms of a fixed point, the focus
$\text{\hspace{0.17em}}P(r,\theta )\text{\hspace{0.17em}}$ at the pole, and a line, the directrix, which is perpendicular to the polar axis.
A conic is the set of all points
$\text{\hspace{0.17em}}e=\frac{PF}{PD},$ where eccentricity
$\text{\hspace{0.17em}}e\text{\hspace{0.17em}}$ is a positive real number. Each conic may be written in terms of its polar equation. See
[link] .
The polar equations of conics can be graphed. See
[link] ,
[link] , and
[link] .
Conics can be defined in terms of a focus, a directrix, and eccentricity. See
[link] and
[link] .
We can use the identities
$\text{\hspace{0.17em}}r=\sqrt{{x}^{2}+{y}^{2}},x=r\text{}\mathrm{cos}\text{}\theta ,$ and
$\text{\hspace{0.17em}}y=r\text{}\mathrm{sin}\text{}\theta \text{\hspace{0.17em}}$ to convert the equation for a conic from polar to rectangular form. See
[link] .
Section exercises
Verbal
Explain how eccentricity determines which conic section is given.
If eccentricity is less than 1, it is an ellipse. If eccentricity is equal to 1, it is a parabola. If eccentricity is greater than 1, it is a hyperbola.
If a conic section is written as a polar equation, and the denominator involves
$\text{\hspace{0.17em}}\mathrm{sin}\text{}\theta ,$ what conclusion can be drawn about the directrix?
Parabola with
$\text{\hspace{0.17em}}e=1\text{\hspace{0.17em}}$ and directrix
$\text{\hspace{0.17em}}\frac{3}{4}\text{\hspace{0.17em}}$ units below the pole.
Hyperbola with
$\text{\hspace{0.17em}}e=2\text{\hspace{0.17em}}$ and directrix
$\text{\hspace{0.17em}}\frac{5}{2}\text{\hspace{0.17em}}$ units above the pole.
Parabola with
$\text{\hspace{0.17em}}e=1\text{\hspace{0.17em}}$ and directrix
$\text{\hspace{0.17em}}\frac{3}{10}\text{\hspace{0.17em}}$ units to the right of the pole.
Ellipse with
$\text{\hspace{0.17em}}e=\frac{2}{7}\text{\hspace{0.17em}}$ and directrix
$\text{\hspace{0.17em}}2\text{\hspace{0.17em}}$ units to the right of the pole.
Hyperbola with
$\text{\hspace{0.17em}}e=\frac{5}{3}\text{\hspace{0.17em}}$ and directrix
$\text{\hspace{0.17em}}\frac{11}{5}\text{\hspace{0.17em}}$ units above the pole.
Hyperbola with
$\text{\hspace{0.17em}}e=\frac{8}{7}\text{\hspace{0.17em}}$ and directrix
$\text{\hspace{0.17em}}\frac{7}{8}\text{\hspace{0.17em}}$ units to the right of the pole.
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic.
Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation
of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15)
it's standard equation is x^2 + y^2/16 =1
tell my why is it only x^2? why is there no a^2?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
Is there any rule we can use to get the nth term ?