Find a Cartesian equation for the curve: $r^2 \cos(2\theta) = 36$

$\boxed{?}$

Identify the curve from the following options:

• Limaçon
• Circle
• Line
• Hyperbola
• Ellipse

Neetesh Kumar | January 2, 2025

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This is the solution to Math 1c
Assignment: 10.3 Question Number 5
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Step-by-step solution:

Step 1: Use trigonometric identity for $\cos(2\theta)$

The given equation is:

$r^2 \cos(2\theta) = 36$

Recall the trigonometric identity:

$\cos(2\theta) = \frac{x^2 - y^2}{x^2 + y^2}$

Also, note that $r^2 = x^2 + y^2$. Substitute these into the given equation:

$(x^2 + y^2) \cdot \frac{x^2 - y^2}{x^2 + y^2} = 36$

Simplify:

$x^2 - y^2 = 36$

Step 2: Identify the curve

The equation $x^2 - y^2 = 36$ represents a hyperbola.

Final Answer:

The Cartesian equation of the curve is:

$\boxed{x^2 - y^2 = 36}$

The curve is a $\boxed{\text{hyperbola}}$

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