Problem 11 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the point ( \(2 \cos \theta, 2 \sin \theta\) ) does not fall in that angle between the lines \(y=|x-2|\) in which the origin lies then \(\theta\) belongs to (A) \(\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)\) (B) \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) (C) \((0, \pi)\) (D) none of these

Step-by-Step Solution

Answer

The angle \(\theta\) belongs to \(\left(\frac{\pi}{2}, \frac{3\pi}{2}\right)\), which is option (A).

Step 1: Understand the Given Condition

The line equations given are \( y = |x - 2| \). These equations describe two lines: the first one is \( y = x - 2 \) and the second one is \( y = -(x - 2) \). The angle between these lines within which the origin lies is between these two lines.

Step 2: Determine Coordinate Boundary

The point \((2 \cos \theta, 2 \sin \theta)\) represents a point on the circle with radius 2. To not fall into the angle where the origin is made by these lines, determine the angles these lines create from the origin. For line \( y = x - 2 \), rearrange to \( y = x - 0 \) or \( y = x \). The angle \( \theta \) for \( y = x \) is \( \frac{\pi}{4} \). For \( y = -(x - 2) \), rearrange to \( y = -x \). The angle for this line is \( \theta = \frac{3\pi}{4} \). The region outside these lines, from the origin, lies between \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{3\pi}{4} \).

Step 3: Identify Complementary Range

Outside this region, the angle \( \theta \) cannot be within \( \left(\frac{\pi}{4}, \frac{3\pi}{4}\right) \). Thus, \( \theta \) will belong to the complementary angle range \( \left(-\infty, \frac{\pi}{4}\right) \cup \left(\frac{3\pi}{4}, \infty\right) \). Simplifying further with trigonometric circles, \( \theta \) is not within this range effectively found in option \( (A) \left(\frac{\pi}{2}, \frac{3\pi}{2}\right) \).

Step 4: Select the Correct Option

From the analysis in step 3, the angle \( \left(\frac{\pi}{2}, \frac{3\pi}{2}\right) \) ensures the point \((2 \cos \theta, 2 \sin \theta)\) is outside the specific region where the origin lies within the lines defined by the absolute function, which matches option (A).

Key Concepts

Coordinate Geometry

Coordinate geometry establishes a link between algebra and geometry through graphs. In the given exercise, we need to examine a point on a circular path. Typically, a point is represented as \(x, y\) on the coordinate plane. Here, the point is given in trigonometric terms as \(2 \cos \theta, 2 \sin \theta\), which signifies that it lies on a circle centered at the origin with a radius of 2.
Coordinate geometry helps us visualize these points on a graph and understand their spatial relations. To assess if the point falls within a specific angular region bounded by certain lines, we use algebraic manipulation and the geometry of lines.
Using the relationships between coordinates and geometric figures is crucial in understanding how physical spaces and angles correspond to numeric values and trigonometric functions.

Circle

A circle is defined as all points equidistant from a center point. In this exercise, the circle's center is at the origin \(0,0\) and has a radius of 2. The point \(2 \cos \theta, 2 \sin \theta\) implies it lies on this circle.

• The cosine and sine functions give the horizontal and vertical positions of a point on the circle.
• The radius of the circle is the distance from the center to any point on its circumference.

The nature of circular motion or points on a circle is intuitive when using trigonometric identities. It allows for straightforward interpretation of these points in both a graphical and algebraic manner.
The circle's role in this exercise is to provide a geometric context where we can apply angle constraints to positions on the circumference.

Angle Measurement

Angles are crucial in understanding the position of points with respect to lines and are measured in radians or degrees. In this problem, angles are analyzed in radians, where \(\pi\) represents half of a circle or 180 degrees.

• Understanding the basic trigonometric circle helps interpret angles between different points and lines.
• Angles are measured from the positive x-axis in the counterclockwise direction.

The lines \(y = x\) and \(y = -x\) correspond to angles \left(\frac{\pi}{4} \right)\ and \left(\frac{3\pi}{4} \right)\, defining the critical separation in this scenario.
These angles assist in defining boundary regions where the point \(2 \cos \theta, 2 \sin \theta\) can lie, outlining spaces divided by certain angular parameters.

Vectors

Vectors represent quantities that have both magnitude and direction. While not explicitly mentioned in the exercise, they play a role in understanding trigonometric movements and direction.
Vectors can be visualized as arrows on a graph, where the direction is given by an angle \(\theta\) and their length by the magnitude, which here is the radius of the circle (2 units).

• The expression \(2 \cos \theta, 2 \sin \theta\) can be seen as a vector originating from (0,0) to a point on the circle.
• Analyzing such vectors provides insights into rotational movements and alignments relative to given lines.

Understanding vectors helps clarify how the orientation and relative positions are considered within the constraints of angles and lines, providing clarity beyond basic point plotting.