find the polar equation for hyperbola eccentricity of .8 and a vertex of (1,pi/2)?
To find the polar equation of a hyperbola with an eccentricity of 0.8 and a vertex at (1, π/2), we need to use the following formula:
r = (e * a) / (1 ± e * cos(θ))
Where:
- r is the distance from the origin to a point on the hyperbola
- e is the eccentricity
- a is the distance from the origin to the center of the hyperbola
- θ is the angle made by the distance from the origin to a point on the hyperbola with the positive x-axis
In this case, we are given that the vertex of the hyperbola is at (1, π/2). This means that the distance from the origin to the center of the hyperbola, a, is 1.
Now, we can substitute the given values into the formula:
r = (0.8 * 1) / (1 ± 0.8 * cos(θ))
This equation gives us the polar equation for the hyperbola with the given eccentricity and vertex. However, note that we have two options for the ± sign in the formula. The positive sign corresponds to the right branch of the hyperbola, while the negative sign corresponds to the left branch.
If you want to focus on the right branch, you can use the positive sign. On the other hand, if you want to focus on the left branch, you can use the negative sign.
So, the polar equation for the hyperbola with an eccentricity of 0.8 and a vertex at (1, π/2) can be expressed as:
For the right branch: r = (0.8 * 1) / (1 + 0.8 * cos(θ))
For the left branch: r = (0.8 * 1) / (1 - 0.8 * cos(θ))