Find a Cartesian equation for the curve: $r = 9 \tan(\theta) \sec(\theta)$

$\boxed{?}$

Identify the curve from the following options:

• Parabola
• Ellipse
• Limaçon
• Line
• Circle

Neetesh Kumar | January 2, 2025

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This is the solution to Math 1c
Assignment: 10.3 Question Number 6
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Step-by-step solution:

Step 1: Express $r$ in terms of Cartesian coordinates:

The relationships between polar and Cartesian coordinates are:

1. $x = r\cos(\theta)$
2. $y = r\sin(\theta)$
3. $\tan(\theta) = \frac{y}{x}$
4. $\sec(\theta) = \frac{1}{\cos(\theta)}$

Given:

$r = 9 \tan(\theta) \sec(\theta)$

Substitute $\tan(\theta) = \frac{y}{x}$ and $\sec(\theta) = \frac{1}{\cos(\theta)}$:

$r = 9 \cdot \frac{y}{x} \cdot \frac{1}{\cos(\theta)}$

Now multiply both sides by $r\cos(\theta)$ to eliminate $r$ and $\cos(\theta)$:

$r \cos(\theta) = 9 \frac{y}{x}$

From $r \cos(\theta) = x$, substitute $x$:

$x = 9 \frac{y}{x}$

Step 2: Simplify the equation:

Multiply through by $x$ to get rid of the denominator:

$x^2 = 9y$

Step 3: Identify the curve:

The equation $x^2 = 9y$ represents a parabola.

Final Answer:

The Cartesian equation of the curve is:

$\boxed{x^2 = 9y}$

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

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