First, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 2, which is
$\text{\hspace{0.17em}}\frac{1}{2}.$
Because
$\text{\hspace{0.17em}}e=\frac{3}{2},e>1,$ so we will graph a
hyperbola with a focus at the origin. The function has a
$\text{\hspace{0.17em}}\mathrm{sin}\text{}\theta \text{\hspace{0.17em}}$ term and there is a subtraction sign in the denominator, so the directrix is
$\text{\hspace{0.17em}}y=-p.$
First, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 5, which is
$\text{\hspace{0.17em}}\frac{1}{5}.$
Because
$\text{\hspace{0.17em}}e=\frac{4}{5},e<1,$ so we will graph an
ellipse with a
focus at the origin. The function has a
$\text{\hspace{0.17em}}\text{cos}\text{\hspace{0.17em}}\theta ,$ and there is a subtraction sign in the denominator, so the
directrix is
$\text{\hspace{0.17em}}x=-p.$
Deﬁning conics in terms of a focus and a directrix
So far we have been using polar equations of conics to describe and graph the curve. Now we will work in reverse; we will use information about the origin, eccentricity, and directrix to determine the polar equation.
Given the focus, eccentricity, and directrix of a conic, determine the polar equation.
Determine whether the directrix is horizontal or vertical. If the directrix is given in terms of
$\text{\hspace{0.17em}}y,$ we use the general polar form in terms of sine. If the directrix is given in terms of
$\text{\hspace{0.17em}}x,$ we use the general polar form in terms of cosine.
Determine the sign in the denominator. If
$\text{\hspace{0.17em}}p<0,$ use subtraction. If
$\text{\hspace{0.17em}}p>0,$ use addition.
Write the coefficient of the trigonometric function as the given eccentricity.
Write the absolute value of
$\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ in the numerator, and simplify the equation.
Finding the polar form of a vertical conic given a focus at the origin and the eccentricity and directrix
Find the polar form of the
conic given a
focus at the origin,
$\text{\hspace{0.17em}}e=3\text{\hspace{0.17em}}$ and
directrix$\text{\hspace{0.17em}}y=-2.$
The directrix is
$\text{\hspace{0.17em}}y=-p,$ so we know the trigonometric function in the denominator is sine.
Because
$\text{\hspace{0.17em}}y=\mathrm{-2},\mathrm{\u20132}<0,$ so we know there is a subtraction sign in the denominator. We use the standard form of
Finding the polar form of a horizontal conic given a focus at the origin and the eccentricity and directrix
Find the
polar form of a conic given a
focus at the origin,
$\text{\hspace{0.17em}}e=\frac{3}{5},$ and
directrix$\text{\hspace{0.17em}}x=4.$
Because the directrix is
$\text{\hspace{0.17em}}x=p,$ we know the function in the denominator is cosine. Because
$\text{\hspace{0.17em}}x=4,4>0,$ so we know there is an addition sign in the denominator. We use the standard form of
Period =2π
if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
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