Question

Identify the curve by finding the Cartesian Equation:
r = 4sec(theta)

so the formulas that can be used here are:
x = rcos(theta)
y = rsin(theta)
x^2 + y^2 = r^2
tan(theta) = y/x

I need to get it to be in the form x^2 + y^2 = r^2

I did:
r = 4sec(theta)
r = 4(r/x)
x = 4
but now... I'm stuck because no matter what I do with that x = 4, I'm don't end up with x^2 + y^2 = (number) format.

Any help would be appreciated. Ty.

Answer 1

\[r=4 \frac{r}{x} =>rx=4r => x=4\]
That is right.

Answer 2

If we had \[r=4\cos(\theta)\]
\[r=4 \frac{x}{r} => r^2=4x => x^2+y^2=4x \]

Answer 3

or even \[r=4\sin(\theta) \]
\[r=4 \cdot \frac{y}{r} => r^2=4y => x^2+y^2=4y \]

Answer 4

not sure if I am correct.
\[r ^{2}=16\sec ^{2}\theta =16\left( 1+\tan ^{2}\theta \right)=16\left( 1+\frac{ y ^{2} }{x ^{2} } \right)\]
\[x ^{2}+y ^{2}=16\left( \frac{ x ^{2}+y ^{2} }{x ^{2} } \right)\]
solve it.

Answer 5

But once I get x=4, how do i convert that into an partesian equation?

Answer 6

x=4 is just a vertical line going through x=4 on the x-axis.