Help with hyperbolic coordinates please B In polar coordinat

B. In polar coordinates, the equation r a is a circle of radius a in the ry-plane. Show that in hyperbolic coordinates, the equation R a is the equation of a hyperbola in ry-plane. Hint: If you are not familiar with the equation of a hyperbola in Cartesian coordinates, you may want to read Section 11.4 in your textbook!

Solution

In polar coordinates, every point of the plane is represented by its distance r from the origin and its angle from the polar axis.

The line segment connecting the two vertices, which lies on the axis, is called the transverse axis, and has length 2a.

Its midpoint is the center of the hyperbola. Perpendicular to the transverse axis at the midpoint is the conjugate axis, whose length is 2b.

The distance from the center to either of the two directrices is a/e. The length of the latus rectum is 2b²/a.

The absolute value of the difference of the distances from any point on the hyperbola to the two foci is 2a.

The equation of the hyperbola has one of the following forms:

                   Standard form:

r²(b²cos²[theta]-a²sin²[theta]) = a²b²,
r = ab/sqrt(-a²+[a²+b²]cos²[theta]),
r = a sqrt(e²-1)/sqrt(-1+e²cos²[theta]).

The center is at the pole O, the foci have coordinates (ae,0) and (ae,Pi), the vertices have coordinates (a,0) and (a,Pi), and the directrices have equations r = ±(a/e)sec(theta). The asymptotes have equations tan(theta) = ±b/a. The tangent at point P₂ has equation

r = a²b²/(a²r₂sin[theta]sin[theta₂]-b²r₂cos[theta]cos[theta₂])

Asymptotic form:

r² = a²/sin( theta).

r²=a²

r=a or R=a

In this form, the center of circle is the pole O, ,and the asymptotes have equations theta = 0 and theta = Pi/2.