First, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 2, which is
Because
so we will graph a
hyperbola with a focus at the origin. The function has a
term and there is a subtraction sign in the denominator, so the directrix is
The directrix is
Plotting a few key points as in
[link] will enable us to see the vertices. See
[link] .
First, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 5, which is
Because
so we will graph an
ellipse with a
focus at the origin. The function has a
and there is a subtraction sign in the denominator, so the
directrix is
The directrix is
Plotting a few key points as in
[link] will enable us to see the vertices. See
[link] .
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
we use the general polar form in terms of sine. If the directrix is given in terms of
we use the general polar form in terms of cosine.
Determine the sign in the denominator. If
use subtraction. If
use addition.
Write the coefficient of the trigonometric function as the given eccentricity.
Write the absolute value of
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,
and
directrix
The directrix is
so we know the trigonometric function in the denominator is sine.
Because
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,
and
directrix
Because the directrix is
we know the function in the denominator is cosine. Because
so we know there is an addition sign in the denominator. We use the standard form of
Abiotic factors are non living components of ecosystem.These include physical and chemical elements like temperature,light,water,soil,air quality and oxygen etc