what is the formula for finding the eccentricity of r=4/-2-costheta and can you explain how to use it?
To find the eccentricity of a polar equation, such as r = 4 / (-2 - cosθ), we can follow a few steps:
Step 1: Rewrite the equation in standard polar form.
Step 2: Identify the coefficients of r and θ.
Step 3: Use these coefficients to calculate the eccentricity.
Let's break down each step in detail:
Step 1: Rewrite the equation in standard polar form.
The given equation is in an alternative polar form called the polar equation. To convert it into standard polar form, we need to eliminate the fraction. Multiply both sides of the equation by (-2 - cosθ) to get:
r * (-2 - cosθ) = 4
Expand the equation to get:
-2r - r * cosθ = 4
Now, rearrange the equation to obtain the standard polar form:
r * cosθ + 2r = -4
Step 2: Identify the coefficients of r and θ.
The equation can be rewritten as:
r(1 * cosθ + 2) = -4
From this, we can identify that the coefficient of r is (1 * cosθ + 2).
Step 3: Calculate the eccentricity.
The eccentricity (ε) of a polar equation is given by the formula:
ε = √(1 + (b² / a²))
In this formula, a represents the coefficient of r, and b represents the coefficient of θ.
In our equation, a = (1 * cosθ + 2), and b = 0 since there is no θ term present.
Therefore, the eccentricity can be calculated as follows:
ε = √(1 + (0² / (1 * cosθ + 2)²))
Simplifying further:
ε = √(1 + 0) = √1 = 1
Hence, the eccentricity of the given polar equation is 1.
Using this value, we can determine the shape of the graph related to this polar equation. An eccentricity of 1 signifies a conic section, specifically an ellipse.