Find a Cartesian equation for the curve: $r = 5\cos(\theta)$

$\boxed{?}$

Identify the curve from the following options:

• Circle
• Limaçon
• Line
• Ellipse
• Hyperbola

Neetesh Kumar | January 2, 2025

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This is the solution to Math 1c
Assignment: 10.3 Question Number 4
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Step-by-step solution:

Step 1: Convert the polar equation to Cartesian form:

The relationships between polar and Cartesian coordinates are:

1. $x = r\cos(\theta)$
2. $y = r\sin(\theta)$
3. $r^2 = x^2 + y^2$

The given equation is $r = 5\cos(\theta)$.

Multiply both sides by $r$ to eliminate $r$ from the denominator:

$r^2 = 5r\cos(\theta)$

Using $r^2 = x^2 + y^2$ and $r\cos(\theta) = x$, substitute these into the equation:

$x^2 + y^2 = 5x$

Step 2: Simplify the equation:

Rearrange the terms:

$x^2 + y^2 - 5x = 0$

Complete the square for the $x$ terms. Add and subtract $\left(\frac{5}{2}\right)^2 = \frac{25}{4}$:

$x^2 - 5x + \frac{25}{4} + y^2 = \frac{25}{4}$

Simplify:

$\left(x - \frac{5}{2}\right)^2 + y^2 = \frac{25}{4}$

Step 3: Identify the curve:

The equation $\left(x - \frac{5}{2}\right)^2 + y^2 = \frac{25}{4}$ represents a circle with:

• Center at $\left(\frac{5}{2}, 0\right)$
• Radius $\frac{5}{2}$

Thus, the curve is a circle.

Final Answer:

The Cartesian equation of the curve is:

$\boxed{x^2 + y^2 - 5x = 0}$

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

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