MHB Convert polar equation sec(theta)=2 to rectangular equation

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To convert the polar equation sec(theta) = 2 to a rectangular equation, start by recognizing that sec(theta) is the reciprocal of cos(theta), leading to cos(theta) = 1/2. By using the relationships x = r*cos(theta) and r = √(x²+y²), multiply both sides by r to derive x = r/2. This results in the equation 4x² = x² + y², which simplifies to y² - 3x² = 0. Understanding these steps can clarify the conversion process from polar to rectangular coordinates.
Elissa89
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My professor gave us a study guide with the solutions:

The equation is:

sec(theta)=2

I am supposed to convert it to a rectangular equation. I know the answer is going to be y^2-3(x)^2=0

I don't know how to get to the answer he gave us.
 
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Elissa89 said:
My professor gave us a study guide with the solutions:

The equation is:

sec(theta)=2

I am supposed to convert it to a rectangular equation. I know the answer is going to be y^2-3(x)^2=0

I don't know how to get to the answer he gave us.

note that $x = r\cos{\theta}$ and $r=\sqrt{x^2+y^2}$ ...$\sec{\theta} = \dfrac{1}{\cos{\theta}} = 2 \implies \cos{\theta}=\dfrac{1}{2}$

multiply both sides by $r$ ...

$r\cos{\theta} = \dfrac{r}{2} \implies x = \dfrac{\sqrt{x^2+y^2}}{2} \implies x^2 = \dfrac{x^2+y^2}{4}$$4x^2 = x^2+y^2 \implies y^2 - 3x^2 = 0$
 
skeeter said:
note that $x = r\cos{\theta}$ and $r=\sqrt{x^2+y^2}$ ...$\sec{\theta} = \dfrac{1}{\cos{\theta}} = 2 \implies \cos{\theta}=\dfrac{1}{2}$

multiply both sides by $r$ ...

$r\cos{\theta} = \dfrac{r}{2} \implies x = \dfrac{\sqrt{x^2+y^2}}{2} \implies x^2 = \dfrac{x^2+y^2}{4}$$4x^2 = x^2+y^2 \implies y^2 - 3x^2 = 0$

*Sigh* It's always so obvious when someone shows me the steps but I have the hardest time figuring it out on my own. thanks
 
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