Convert hyperbola in rectangular form to polar form

rhedynogh0rp 2022-03-25 Answered
Convert hyperbola in rectangular form to polar form
3y216yx2+16=0.
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Answers (1)

Mason Knight
Answered 2022-03-26 Author has 11 answers

Step 1
Staring with
3y216yx2+16=0
Put it first in the standard format. Compute the square in y
3(yd83)2d643x2+16=0
3(yd83)2x2=d163
d916(yd83)2d316x2=1
Thus the center is at
(0, 83), a=43 and b=43
The focal distance
c=a2+b2=4d19+d13=d83
Hence, the foci are at (0,0) and (0,163)
Taking the focus that is at the origin, then x=rcosθ, y=rsinθ
Back to the original equation
3y216yx2+16=0
Substituting the polar expressions,
3(rsinθ)216rsinθr2cos2θ+16=0
Using cos2θ=1sin2θ and collecting terms
r2(4sin2θ1)16rsinθ+16=0
From the quadratic formula, and taking the positive root
r=d12(4sin2θ1) (16sinθ256sin2θ(256sin2θ64))
And this simplifies to
r=d12(4sin2θ1)(16sinθ8)
Factoring the denominator and cancelling equal terms
r=d42sinθ+1

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