Question

find the polar subtangents of the curves
r=2a/1-costheta

Answer 1

Answer:

Given 2a/ r = (1 - cosθ)

Taking log on both sides,

log2a = log r + log(1 – cosθ)

On differentiation,  0 = 1/r . dr/dθ + sinθ/ 1 − cosθ

or    1/r dr/dθ = - cotθ/2  

dθ/dr = - (tan θ/2)/ r  

Therefore    tan φ = rdθ/dr = r( - (tan θ/2)/ r) = - tanθ/2 =  tan ( π - θ/2)

implying  φ = π − θ/2

Again, we know that, p = r sin φ = r sin (π - θ/2) = 2a/ (1 - cos θ) . sin θ/2  

= 2a/ (2sin2) θ/2 . sin θ/2

implying p = a cosecθ/2

or  

For polar subtangent, OT  

r tan φ = 2a/ (1 - cos θ) . tan(π - θ/2)

Step-by-step explanation: