Given: r = 4/-2-6sintheta
What is the eccentricity of the function?
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2
-2
3
-3
To find the eccentricity of the function, we need to determine if it is an ellipse, hyperbola, or parabola, and then use the properties of that shape to calculate the eccentricity.
The given equation is in the form r = a + b*sin(theta), where r is the distance from the origin to a point on the curve, and theta is the angle that line from the origin to the point makes with the horizontal axis.
To identify the shape, let's consider the possible values of a and b in the equation r = a + b*sin(theta).
If a > |b|, the shape is an ellipse.
If a = |b|, the shape is a parabola.
If a < |b|, the shape is a hyperbola.
In our equation, r = 4/-2-6*sin(theta), a = 4, and b = -6.
Since a = 4 > |-6| = 6, the shape is an ellipse.
The eccentricity of an ellipse is given by the formula e = sqrt(1 - (b^2)/(a^2)).
Plugging in the values, we have:
e = sqrt(1 - ((-6)^2)/(4^2))
e = sqrt(1 - 36/16)
e = sqrt(1 - 9/4)
e = sqrt(4/4 - 9/4)
e = sqrt(-5/4)
Since the square root of a negative number is not a real number, we can conclude that the eccentricity of the function is not a real number. Therefore, none of the given options (2, -2, 3, -3) are the correct answer.