Question

Sketch the graph of the given equation and find the area of the region bounded by it. $$ r=2 a \cos \theta, a>0 $$

Short answer

The graph is a circle centered at \((a, 0)\) with radius \(a\), having area \(\pi a^2\).

Step-by-step solution

Understanding the Equation

The given equation is in the polar form: \( r = 2a \cos \theta \). It represents a circle with its center at \( (a, 0) \) and radius \( a \). Polar coordinates \( r \) and \( \theta \) describe the position of points on this circle relative to the origin.

Sketching the Graph

To sketch this graph, consider plotting key points for different values of \( \theta \). When \( \theta = 0 \), \( r = 2a \), placing a point at \( (2a, 0) \). When \( \theta = \frac{\pi}{2} \) or \( \theta = -\frac{\pi}{2} \), \( r = 0 \), placing these points at the origin. When \( \theta = \pi \), \( r = -2a \), equivalent to \( (2a, 0) \). The graph is a circle centered at \( (a, 0) \) with radius \( a \).

Finding the Area of the Circle

The area of a circle is given by the formula \( \pi \times \text{radius}^2 \). In our circle, the radius is \( a \). Thus, the area is \( \pi \times a^2 \).

Conclusion

The graph of the equation \( r = 2a \cos \theta \) describes a circle of radius \( a \) centered at \( (a, 0) \). The bounded area is \( \pi a^2 \).

Key concepts

Circle Equation

In the world of polar coordinates, the equation \( r = 2a \cos \theta \) is a fascinating one because it represents a circle. Unlike the usual Cartesian equations you might be familiar with, polar equations describe the location of points using a distance \( r \) from the origin and an angle \( \theta \) from the positive x-axis.

For this specific equation, \( r = 2a \cos \theta \), it describes a circle with its center not at the origin but at \((a, 0)\) in the Cartesian plane. The term \(2a \cos \theta\) tells us that as \( \theta \) varies, the radial distance changes according to the value of the cosine function. This equation is symmetric about the polar axis (the positive x-axis), reflecting how cosine functions behave.

The appearance of \( 2a \) conveys vital information about the geometry of the circle. It is twice the x-coordinate of the circle's center \((a,0)\). Learning to interpret polar equations like this opens up a deeper understanding of geometry, showing that coordinate systems can offer unique perspectives based on their symmetries and transformations.

Area Calculation

Calculating the area enclosed by a circle in polar coordinates is straightforward once you know the formula for the area of a circle: \( \pi \times \text{radius}^2 \).

With the given equation \( r = 2a \cos \theta \), we already established that the radius of this circle is \( a \). Thus, the total area of the circle, which is the space within its boundary, is given by \( \pi \times a^2 \).

This calculation leverages the knowledge of geometry that the total area of a circle doesn’t depend on its position in the coordinate system, whether Cartesian or polar. The key here is acknowledging the shift in the circle’s center and using the identified radius to perform the straightforward area calculation.

Understanding these calculations allows you to tackle a wide variety of geometric problems efficiently, viewing them through the lens of different mathematical perspectives.

Graph Sketching

Graph sketching in polar coordinates starts with understanding how the radius \( r \) changes as the angle \( \theta \) varies. For the circle equation \( r = 2a \cos \theta \), you can plot the circle by tracking changes in \( r \) for notable angles.

Here’s a simple way to approach this:

• For \( \theta = 0 \), \( r = 2a \cdot \cos(0) = 2a \), placing a point farthest to the right at \((2a, 0)\).
• At \( \theta = \frac{\pi}{2} \) or \( \theta = -\frac{\pi}{2} \), \( r = 0 \), indicating points lie at the origin.
• For \( \theta = \pi \), \( r = 2a \cdot \cos(\pi) = -2a \), which graphically also places the point back at \((2a, 0)\), but visually connects back in an opposite manner.

This creates a complete circle centered at \((a, 0)\) with radius \( a \). By visualizing these points, the symmetry in the function makes it easier to draw the full circle smoothly.

Learning to sketch graphs this way can broaden your comprehension of how different coordinate systems transform and represent shapes and how symmetry plays a crucial role in graph representation.