Provide the standard equations for lines and conic sections in polar coordinates with examples.

Question

Gennenzip · Accepted Answer

The standard equation for lines in polar coordinates.
If the

P_{0} (r_{0}, θ_{0})

is the foot of the perpendicular from the origin to the line L, and

r_{0} \geq 0

, then an equation for L is

\cos (θ - θ_{0}) = r_{0}

.
Example:
Consider

θ_{0} = \frac{π}{3}

and

r_{0} = 2 .

The equation for line is

r \cos (θ - θ_{0}) = r_{0}

Substitute

\frac{π}{3}

for

θ_{0}

and 2 for

r_{0}

r \cos (θ - \frac{π}{3}) = 2

r (\cos θ \cos \frac{π}{3} + \sin θ \sin \frac{π}{3}) = 2

\frac{1}{2} r \cos θ + \frac{\sqrt{3}}{2} r \sin θ = 2

r \cos θ + \sqrt{3} r \sin θ = 4

Substitute x for

r \cos θ

and y for

r \sin θ

x + \sqrt{3 y} = 4

Thus, the equation of line is

x + \sqrt{3 y} = 4

The equation for a conic with eccentricity e is

r = \frac{k e}{1 + e \cos θ}

where,

x = k > 0

is the vertical directrix.
Example:
Consider the equation of hyperbola with the eccentricity

\frac{3}{2}

and directrix

x = 2

The equation for a conic with eccentricity e is

r = \frac{k e}{1 + e \cos θ}

Substitute 2 for k and

\frac{3}{2}

for e.

r = \frac{2 \times \frac{3}{2}}{1 + \frac{3}{2} \cos θ}

= \frac{3}{\frac{2 + 3 \cos θ}{2}}

= \frac{6}{2 + 3 \cos θ}

Hence, the equation for a conic is

r = \frac{6}{2 + 3 \cos θ}

Provide the standard equations for lines and conic sections in