beljuA
2021-08-11
Answered

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

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Gennenzip

Answered 2021-08-12
Author has **96** answers

The standard equation for lines in polar coordinates.

If the${P}_{0}({r}_{0},\text{}{\theta}_{0})$ is the foot of the perpendicular from the origin to the line L, and ${r}_{0}\ge 0$ , then an equation for L is $\mathrm{cos}(\theta -{\theta}_{0})={r}_{0}$ .

Example:

Consider$\theta}_{0}=\frac{\pi}{3$ and ${r}_{0}=2.$

The equation for line is$r\mathrm{cos}(\theta -{\theta}_{0})={r}_{0}$

Substitute$\frac{\pi}{3}$ for $\theta}_{0$ and 2 for $r}_{0$

$r\mathrm{cos}(\theta -\frac{\pi}{3})=2$

$r(\mathrm{cos}\theta \mathrm{cos}\frac{\pi}{3}+\mathrm{sin}\theta \mathrm{sin}\frac{\pi}{3})=2$

$\frac{1}{2}r\mathrm{cos}\theta +\frac{\sqrt{3}}{2}r\mathrm{sin}\theta =2$

$r\mathrm{cos}\theta +\sqrt{3}r\mathrm{sin}\theta =4$

Substitute x for$r\mathrm{cos}\theta$ and y for $r\mathrm{sin}\theta$

$x+\sqrt{3y}=4$

Thus, the equation of line is$x+\sqrt{3y}=4$

The equation for a conic with eccentricity e is

$r=\frac{ke}{1+e\mathrm{cos}\theta}$

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$r=\frac{ke}{1+e\mathrm{cos}\theta}$

Substitute 2 for k and$\frac{3}{2}$ for e.

$r=\frac{2\times \frac{3}{2}}{1+\frac{3}{2}\mathrm{cos}\theta}$

$=\frac{3}{\frac{2+3\mathrm{cos}\theta}{2}}$

$=\frac{6}{2+3\mathrm{cos}\theta}$

Hence, the equation for a conic is$r=\frac{6}{2+3\mathrm{cos}\theta}$

If the

Example:

Consider

The equation for line is

Substitute

Substitute x for

Thus, the equation of line is

The equation for a conic with eccentricity e is

where,

Example:

Consider the equation of hyperbola with the eccentricity

The equation for a conic with eccentricity e is

Substitute 2 for k and

Hence, the equation for a conic is

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