Question

Using polar equations, derive the formulas for the focal width of an ellipse and the focal width of a hyperbola.Begin by defining focal width for these conics in a manner analogous to the definition of the focal width of a parabola.

Answer 1

i don't get it, why do you need polar equations?
the formula for the focal width is:
ellipe
c = sqrt(a^2 -b^2)

hyperbola
c = sqrt(a^2 +b^2)

Answer 2

i don't why too:/

Answer 3

but can you use polar to solve it please?

Answer 4

I think put, \( \theta\) = 0 or \(\theta = \pi\) will give you r's, multiply it by 2\( e \) , that COULD be your focal width.

Answer 5

polar form for ellipse
\[r = \frac{ab}{\sqrt{a^{2}\sin^{2} \theta + b^{2}\cos^{2} \theta}}\]

Answer 6

um

Answer 7

what about the directed angle?

Answer 8

not sure what you mean

Answer 9

theta

Answer 10

you told me how to get r but how would i get theta?

Answer 11

? polar equations are usually a function of theta ....r = f(theta)
anyway from there i don't know how to derive the focal width c
sorry

Answer 12

well thanks for all your help dumbcow andexperimentX

Answer 13

If the ellipse is standard ellipse centered on origin, then obviously, theta will be zero or pi

Answer 14

when theta = 0 or pi
r = a

Answer 15

Ah .. yes, 2ea <--- that will be our focal width. <--- should have thought earlier.