Problem 91 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The equation \(r=|\cos \theta|\) represents (A) two circles of radii \(\frac{1}{2}\) each (B) two circles centered at \(\left(\frac{1}{2}, 0\right)\) and \(\left(-\frac{1}{2}, 0\right)\) (C) two circles touching each other at the origin (D) pair of straight lines

Step-by-Step Solution

Answer

The equation represents (C) two circles touching each other at the origin.

Step 1: Understanding the Polar Equation

We are given the polar equation \( r = |\cos \theta| \). In polar coordinates, \( r \) is the distance from the origin (pole), and \( \theta \) is the angle from the positive x-axis. The equation describes the relationship between \( r \) and \( \theta \).

Step 2: Simplifying the Equation

The equation can be broken into two cases due to the absolute value: 1) \( r = \cos \theta \) when \( \cos \theta \geq 0 \), and 2) \( r = -\cos \theta \) when \( \cos \theta < 0 \).

Step 3: Case 1: When \( \cos \theta \geq 0 \)

Here, \( r = \cos \theta \). Converting to Cartesian coordinates using the relations \( x = r \cos \theta \) and \( y = r \sin \theta \), we have \( y = r \sin \theta = \sin \theta \cdot \cos \theta \). Notice, however, that for this part of the graph, \( r = x \) when converting back, leading to the circle equation \( (x - \frac{1}{2})^2 + y^2 = \frac{1}{4} \).

Step 4: Case 2: When \( \cos \theta < 0 \)

In this case, \( r = -\cos \theta \). By similar reasoning, converting to Cartesian results in the circle centered at \( (-\frac{1}{2}, 0) \) with the same radius \( \frac{1}{2} \), or \( (x + \frac{1}{2})^2 + y^2 = \frac{1}{4} \).

Step 5: Analyzing Circle Properties

Both circles obtained from the equations \( (x - \frac{1}{2})^2 + y^2 = \frac{1}{4} \) and \( (x + \frac{1}{2})^2 + y^2 = \frac{1}{4} \) have radius \( \frac{1}{2} \) and touch each other at the origin, which means they satisfy option (C).

Step 6: Conclusion

The equation \( r=|\cos \theta| \) indeed represents two circles with radii \( \frac{1}{2} \) each, centered at \( (\pm \frac{1}{2}, 0) \) and touching at the origin.

Key Concepts

Circle Equations

Circle equations are a fundamental part of coordinate geometry and describe a perfect round shape. In a standard Cartesian coordinate system, a circle is represented by the equation \[(x - h)^2 + (y - k)^2 = r^2\]where:

• \( (h, k) \) is the center of the circle.
• \( r \) is the radius, indicating how far out the circle stretches from its center.

In the given problem, we derived the equations \((x - \frac{1}{2})^2 + y^2 = \frac{1}{4}\) and \((x + \frac{1}{2})^2 + y^2 = \frac{1}{4}\). These tell us that we have two circles.
Each has a radius of \( \frac{1}{2} \) and centers at \( (\frac{1}{2}, 0) \) and \( (-\frac{1}{2}, 0) \), respectively. When graphed, these circles are perfect mirror images along the y-axis and just touch each other at the point \( (0, 0) \), known as the origin.

Absolute Value Function

The absolute value function takes a number and turns it into its non-negative form. It's denoted as \(|x|\) and can be mathematically defined as:

• \(|x| = x\) if \(x \geq 0\)
• \(|x| = -x\) if \(x < 0\)

This function is crucial in the problem since the polar coordinate equation \( r = |\cos \theta| \) is at its heart. Here, \(|\cos \theta|\) splits the problem into two scenarios:
1. When \( \cos \theta \geq 0\), it stays as \( \cos \theta \).2. When \(\cos \theta < 0\), it changes to \(-\cos \theta \).
By handling these differently, it ensures that \( r \) always remains non-negative, making \( r \) a valid distance in polar coordinates. Understanding this keeps the polar equation meaningful and interprets it effectively into Cartesian form.

Conversion Between Polar and Cartesian Coordinates

Converting between polar and Cartesian coordinates is essential when dealing with shapes like circles. Polar coordinates use \((r, \theta)\), where \(r\) is the radial distance and \(\theta\) is the angle from the positive x-axis. Cartesian coordinates, however, use \((x, y)\) to plot a position in a rectangular grid.
The conversion formulas are:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)

In our exercise, we convert the polar equation \(r = |\cos \theta|\) into two Cartesian forms that describe two circles. The use of these conversions allows us to see geometric realities hidden in polar form. We can compare properties like radius and center using conventional Cartesian equations.
Converting helps us move seamlessly between two ways to visualize spatial relationships, enhancing our understanding of geometric figures.