Provide the standard equations for lines and conic sections in

beljuA 2021-08-11 Answered
Provide the standard equations for lines and conic sections in polar coordinates with examples.
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Expert Answer

Gennenzip
Answered 2021-08-12 Author has 96 answers
The standard equation for lines in polar coordinates.
If the P0(r0, θ0) is the foot of the perpendicular from the origin to the line L, and r00, then an equation for L is cos(θθ0)=r0.
Example:
Consider θ0=π3 and r0=2.
The equation for line is rcos(θθ0)=r0
Substitute π3 for θ0 and 2 for r0
rcos(θπ3)=2
r(cosθcosπ3+sinθsinπ3)=2
12rcosθ+32rsinθ=2
rcosθ+3rsinθ=4
Substitute x for rcosθ and y for rsinθ
x+3y=4
Thus, the equation of line is x+3y=4
The equation for a conic with eccentricity e is
r=ke1+ecosθ
where, x=k>0 is the vertical directrix.
Example:
Consider the equation of hyperbola with the eccentricity 32 and directrix x=2
The equation for a conic with eccentricity e is r=ke1+ecosθ
Substitute 2 for k and 32 for e.
r=2×321+32cosθ
=32+3cosθ2
=62+3cosθ
Hence, the equation for a conic is r=62+3cosθ
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