Step by step solution to "Find a Cartesian equation for the curve and..."

ka1leE

Answered question

2021-05-01

Find a Cartesian equation for the curve and identify it.

r^{2} \cos 2 θ = 1

2abehn

Skilled2021-05-02Added 88 answers

Solved in photo:

Jeffrey Jordon

Expert2021-09-30Added 2605 answers

Consider the polar equation,

$R^{2} \cos 2 θ = 1$

Need to convert this equation into Cartesian equation.

Let the rectangular coordinates be $(x, y)$

The polar coordinates $(r, θ)$ is defined as

$R^{2} = x^{2} + y^{2} x = R \cos θ y = R \sin θ$

So that $\tan θ = \frac{y}{x}$

Therefore

$R^{2} \cos 2 θ = 1$

$R^{2} (\cos^{2} θ - \sin^{2} θ) = 1$

$R^{2} \cos^{2} θ - R \sin^{2} θ = 1$

$(R \cos θ)^{2} - (R \sin θ)^{2} = 1$

$x^{2} - y^{2} = 1$

Thus, the cartesian equation is $x^{2} - y^{2} = 1$ and it is a hyperbola

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