r=4/(-2-6sinϴ)
1. What is the eccentricity of the function?
2. What is the distance between the pole and the directrix?
1. A. 2
B. -2
C. 3
D. -3
2. A. 2
B. 2/3
C. 3
D. 6
So, I guess you did not read the web page I suggested ...
r = ed/(1+e sin?)
r=4/(-2-6sin?)
r = -2/(1+3sin?)
so, e=3 and d=2/3
in x-y coordinates, that is
(y-3/4)^2 - x^2/8 = 1/16
http://www.wolframalpha.com/input/?i=hyperbola+(y-3%2F4)%5E2+-+x%5E2%2F8+%3D+1%2F16
You can see that with a focus at (0,0), vertex at (0,1/2) and eccentricity of 3 it agrees with the information from the polar form.
To find the eccentricity of a function in polar coordinates, you can use the formula:
e = sqrt(1 + b^2 / a^2)
where a is the absolute value of the coefficient of the sine term, and b is the coefficient of the cosine term.
In the given equation, r = 4 / (-2 - 6sinϴ), the coefficient of the sine term is -6. Taking the absolute value, we get a = 6. The coefficient of the cosine term is -2. Therefore, b = -2.
Now, substitute the values of a and b into the eccentricity formula:
e = sqrt(1 + (-2)^2 / 6^2)
e = sqrt(1 + 4/36)
e = sqrt(1 + 1/9)
e = sqrt(10/9)
e = sqrt(10) / 3
Thus, the eccentricity of the given function is sqrt(10) / 3. Since none of the given options match this value, the answer is not provided in the given choices.
To find the distance between the pole and the directrix of a polar function, you can use the formula:
D = a / (1 - e)
where a is the absolute value of the coefficient of the cosine term, and e is the eccentricity.
In the given equation, r = 4 / (-2 - 6sinϴ), the coefficient of the cosine term is -2. Taking the absolute value, we get a = 2. The eccentricity, as we calculated earlier, is sqrt(10) / 3.
Now, substitute the values of a and e into the formula:
D = 2 / (1 - sqrt(10) / 3)
To simplify this expression, multiply the numerator and denominator by 3 to eliminate the fraction in the denominator:
D = 6 / (3 - sqrt(10))
Thus, the distance between the pole and the directrix is 6 / (3 - sqrt(10)). Among the given options, the closest match is D. 6.