SOLUTION: Find the polar equation of an ellipse with a focus at the pole and major axis endpoints (4, 0) and (2, pi).

Algebra ->  Equations -> SOLUTION: Find the polar equation of an ellipse with a focus at the pole and major axis endpoints (4, 0) and (2, pi).      Log On


   

Question 1136451: Find the polar equation of an ellipse with a focus at the pole and major axis endpoints (4, 0)
and (2, pi).
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
given: ellipse with a focus at the pole and major axis endpoints (4, 0) and (2, pi).

r+=+ke+%2F+%28+1+-+e+%2Acos+%28theta%29%29


substitute in points:

4+=+ke%2F+%28+1+-+cos+%280%29+%29==> cos+%280%29=1

4+=+ke%2F+%28+1+-+e%281%29+%29

4+=+ke+%2F+%28+1+-+e+%29+

ke=4%28+1+-+e+%29+

ke=4+-+4e

k=4%2Fe-4......eq.1


2+=+ke%2F+%28+1+-cos+%28pi%29%29==> cos+%28pi%29=-1

2+=+ke%2F+%28+1+-+e%28-1%29+%29

2+=+ke%2F+%28+1+%2B+e+%29

ke=2+%2B+2e+

k=2%2Fe%2B2......eq.2


from eq.1 and eq.2 we have

4%2Fe-4=2%2Fe%2B2

4%2Fe-2%2Fe=2%2B4

2%2Fe=6

2=6%2Ae

e=2%2F6

e=1%2F3


find k

k=4%2Fe-4......eq.1

k=4%2F%281%2F3%29-4

k=12-4

k=8


r+=+ke+%2F+%28+1+-+e+%2Acos+%28theta%29%29...substitute in k and e

r+=%288%281%2F3%29%29+%2F+%28+1+-+%281%2F3%29+%2Acos+%28theta%29%29

r+=%288%2F3%29+%2F+%28+1+-+%281%2F3%29+%2Acos+%28theta%29%29......both numerator and denominator multiply by 3

r+=3%288%2F3%29+%2F+%28+3+-+3%281%2F3%29+%2Acos+%28theta%29%29

r+=8+%2F+%28+3+-+cos+%28theta%29%29