Find the values of e,a,b,and c for r=21/5-2costheta?
To find the values of e, a, b, and c in the equation r = 21/5 - 2cos(theta), we can compare it with the standard form of a polar equation:
r = e + a cos(b(theta - c))
Comparing the given equation with the standard form, we can derive the values of e, a, b, and c.
1. The value of e is the distance of the directrix from the pole, which is the constant term in the equation. In the given equation, e = 21/5.
2. The value of a is the length of the latus rectum, which is the coefficient of cos(theta). In the given equation, a = -2.
3. The value of b determines the number of petals or lobes in polar graphs. In the given equation, b is not explicitly mentioned. To find b, we can simplify the equation as much as possible.
Multiply both sides of the equation by 5 to get rid of the fraction:
5r = 21 - 10cos(theta)
Now, rearrange the equation:
10cos(theta) = 21 - 5r
Divide both sides by 10:
cos(theta) = (21 - 5r) / 10
The equation is now in the form of cos(theta) = f(r), where f(r) is a function of r. By comparing it with the standard form cos(theta) = cos(b(theta - c)), we find that b = 1.
4. The value of c determines the orientation of the polar graph. In the given equation, c = 0, since there is no shift in the angle theta.
Therefore, the values of e, a, b, and c in the equation r = 21/5 - 2cos(theta) are:
e = 21/5
a = -2
b = 1
c = 0
I assume you mean
r = 21/(5-2cosθ)
When you consider the general equation for a conic:
r = ep/(1 - e cosθ)
you see that you must set it up as
r = 21/[5(1 - 2/5 cosθ))
Now you can read off e and p. Since e<1, you have an ellipse. Now you need to use the distance to the directrix to determine a; then e will give you b and c.