Convert the polar equation to a rectangular equation.

$r=-9\mathrm{cos}\theta $

$r=-9\mathrm{cos}\theta $

heelallev5
2022-08-05
Answered

Convert the polar equation to a rectangular equation.

$r=-9\mathrm{cos}\theta $

$r=-9\mathrm{cos}\theta $

You can still ask an expert for help

Nicole Soto

Answered 2022-08-06
Author has **10** answers

$r=\sqrt{{x}^{2}+{y}^{2}}$ and $x=r\mathrm{cos}\theta \Rightarrow \mathrm{cos}\theta =\frac{x}{r}$

$\therefore r=-9\mathrm{cos}\theta =-9\ast \frac{x}{r}$

${r}^{2}=-9x$

$\Rightarrow {x}^{2}+{y}^{2}=-9x$

$\Rightarrow (x+\frac{9}{2}{)}^{2}+{y}^{2}=\frac{81}{4}\leftarrow $ Completing the square

$\Rightarrow \frac{(x+\frac{9}{2}{)}^{2}}{(\frac{9}{2}{)}^{2}}+\frac{{y}^{2}}{(\frac{9}{2}{)}^{2}}=1\leftarrow $ Equation of an ellipse

$\therefore r=-9\mathrm{cos}\theta =-9\ast \frac{x}{r}$

${r}^{2}=-9x$

$\Rightarrow {x}^{2}+{y}^{2}=-9x$

$\Rightarrow (x+\frac{9}{2}{)}^{2}+{y}^{2}=\frac{81}{4}\leftarrow $ Completing the square

$\Rightarrow \frac{(x+\frac{9}{2}{)}^{2}}{(\frac{9}{2}{)}^{2}}+\frac{{y}^{2}}{(\frac{9}{2}{)}^{2}}=1\leftarrow $ Equation of an ellipse

ghettoking6q

Answered 2022-08-07
Author has **8** answers

Given the polar equation

$r=-9\mathrm{cos}\theta $

Multiply both sides of the equation by r.

${r}^{2}=-9\mathrm{cos}\theta $

Use the following conversions: ${r}^{2}={x}^{2}+{y}^{2},x=r\mathrm{cos}\theta $.

${x}^{2}+{y}^{2}=-9x$

Add 9x to both sides of the equation.

$({x}^{2}+9x)+{y}^{2}=0$

Complete the square.

$({x}^{2}+9x\frac{81}{4})+{y}^{2}=\frac{81}{4}$

$(x+\frac{9}{2}{)}^{2}+{y}^{2}=\frac{81}{4}$....Rectangular Equation

Conic Section is a Circle

center: (h, k)------(-9/2, 0)

radius (r): 9/2

$r=-9\mathrm{cos}\theta $

Multiply both sides of the equation by r.

${r}^{2}=-9\mathrm{cos}\theta $

Use the following conversions: ${r}^{2}={x}^{2}+{y}^{2},x=r\mathrm{cos}\theta $.

${x}^{2}+{y}^{2}=-9x$

Add 9x to both sides of the equation.

$({x}^{2}+9x)+{y}^{2}=0$

Complete the square.

$({x}^{2}+9x\frac{81}{4})+{y}^{2}=\frac{81}{4}$

$(x+\frac{9}{2}{)}^{2}+{y}^{2}=\frac{81}{4}$....Rectangular Equation

Conic Section is a Circle

center: (h, k)------(-9/2, 0)

radius (r): 9/2

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