- #1
subwaybusker
- 51
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Homework Statement
Show that the parabolas r=c/(1+cosθ)and r'=d/(1-cosθ) intersect at right angles.
The Attempt at a Solution
I found the points of intersection by setting the two equations equal, to which I got:
cosθ = (c- d)/(c+d)
θ = cos^-1[(c- d)/(c+d)]
then i tried to find the slope of the two equations:
x=dcosθ/1-cosθ ; y=dsinθ/1-cosθ
dy/dθ = [dcosθ(1-cosθ)-(sinθ)dsinθ] / (1-cosθ)^2 = d(cosθ-1)/(1-cosθ)^2
dx/dθ= [-dsinθ(1-cosθ)-(sinθ)dcosθ] / (1-cosθ)^2 = -dsinθ/(1-cosθ)^2
dy/dx=cosθ-1/-sinθ
x=ccosθ/1+cosθ ; y=csinθ/1+cosθ
dy'/dθ = [-csinθ(1+cosθ)-(-sinθ)ccosθ] / (1+cosθ)^2 = -csinθ/(1+cosθ)^2
dx'/dθ= [ccosθ(1+cosθ)-(-sinθ)csinθ] / (1+cosθ)^2 = c(cosθ+1)/(1+cosθ)^2
dy/dx=cosθ+1/-sinθ
Then I don't know what to do