Given: r = 4/(-2-6sinθ)
What is the eccentricity of the function?
A. 2
B. -2
C. 3
D. -3
To find the eccentricity of a function, we need to use the equation of the ellipse in standard form:
r = a(1 - e^2) / (1 + e cosθ)
Where r is the distance from the origin to a point (x, y) on the ellipse, a is the length of the semi-major axis, e is the eccentricity, and θ is the angle between the major axis and the line connecting the origin to the point (x, y).
In this case, we are given r = 4/(-2 - 6sinθ). To compare this with the standard form of the ellipse equation, we need to rewrite it in terms of r and sinθ.
Multiply both the numerator and denominator by (-1) to get:
r = -4 / (2 + 6sinθ)
Now we can compare this with the equation of the ellipse in standard form:
r = a(1 - e^2) / (1 + e cosθ)
Comparing the above two equations, we can see that the semi-major axis (a) is equal to -4, and the eccentricity (e) is the unknown. Therefore, we can rewrite the equation as:
-4 = -4(1 - e^2) / (1 + e cosθ)
Simplifying this equation, we get:
1 - e^2 = 1 + e cosθ
We can cancel out the common factors of -4 and 1:
-1 + e^2 = 1 + e cosθ
Moving all the terms to the left side, we get:
e^2 + e cosθ - 2 = 0
Now, using the quadratic formula, we can solve for e:
e = (-cosθ ± √(cosθ^2 - 4(-2))) / 2
Simplifying further, we have:
e = (-cosθ ± √(cosθ^2 + 8)) / 2
Therefore, the eccentricity of the function is given by the equation:
e = (-cosθ ± √(cosθ^2 + 8)) / 2
Since the eccentricity is dependent on the angle θ, there is no single value that represents the eccentricity for all values of θ. Therefore, none of the options A, B, C, or D are the correct answer.