Problem 24 - A Complete Resource Book in Mathematics for JEE Main

Question

The total area enclosed by the lines \(y=|x|, y=0\) and \(|x|=1\) is (A) 2 (B) 4 (C) 1 (D) None of these

Step-by-Step Solution

Answer

The total area enclosed by the lines is 1, so the answer is (C) 1.

Step 1: Understand the lines forming the region

The lines forming the region are given by the equations \(y = |x|, y = 0, \text{ and } |x| = 1.\) The equation \(y = |x|\) forms two lines: \(y = x\) for \(x \geq 0\) and \(y = -x\) for \(x < 0\). The line \(y = 0\) is the x-axis, and \(|x| = 1\) consists of the vertical lines \(x = 1\) and \(x = -1\). The region enclosed by these lines is a triangle with its base along the x-axis ranging from \(-1\) to \(1\).

Step 2: Identify the vertices of the triangle

The intersections of these lines define the vertices of the enclosed region. The vertices are where \(y = x\) intersects \(y = 0\) at \((0,0)\), \(y = -x\) intersects \(y = 0\) at \((0,0)\), \(y = x\) intersects \(x = 1\) at \((1,1)\), and \(y = -x\) intersects \(x = -1\) at \((-1,1)\). Hence, the vertices are \((-1,1), (0,0), (1,1)\).

Step 3: Visualize the enclosed region

This region is a symmetric triangle above the x-axis with its base along the x-axis between points \((-1,0)\) and \((1,0)\) and its top corners at \((-1,1)\) and \((1,1)\).

Step 4: Calculate the area of the triangle

The triangle has a base of length 2 (from \(x = -1\) to \(x = 1\)) and a height of 1 (from \(y = 0\) to the vertex points \((1,1)\) and \((-1,1)\)). The area \(A\) of a triangle is given by the formula \(A = \frac{1}{2} \times \text{base} \times \text{height}\). Thus, the area is \(A = \frac{1}{2} \times 2 \times 1 = 1\).

Key Concepts

Area of a Triangle

Understanding how to calculate the area of a triangle is essential in coordinate geometry. In this problem, a triangle is formed by the lines defined in the coordinate plane. To find its area, we use the formula:

• Area, \(A = \frac{1}{2} \times \text{base} \times \text{height}\).

For this specific triangle, the base is the line segment along the x-axis between \(x = -1\) and \(x = 1\), providing a base length of \(2\).
The height of the triangle is derived from the maximum \(y\)-coordinate of the vertices, which is \(1\).
Applying these values into the area formula results in:

• \(A = \frac{1}{2} \times 2 \times 1 = 1\).

The result here is an area of \(1\) square unit, which confirms the solution in the exercise.

Absolute Value Functions

Absolute value functions can be perplexing at first, but they are vital in understanding many problems involving linear graphing.
The absolute value function \(y = |x|\) can be expressed as two separate lines: \(y = x\) when \(x \geq 0\) and \(y = -x\) when \(x < 0\).
This piecewise nature forms a "V"-shape in the coordinate plane, intersecting the x-axis at the origin, \((0,0)\). Understanding this can help students visualize how absolute value impacts graph shapes.When we consider constraints like \(|x| = 1\), it simplifies to two vertical lines \(x = -1\) and \(x = 1\). These lines bound a region, in this case, helping form the enclosed triangular region.

Coordinate Plane

The coordinate plane is a two-dimensional space where any point can be defined by a pair of numerical coordinates. These coordinates are in the format \((x, y)\), which describe the horizontal and vertical positioning from a central point known as the origin, \((0,0)\).
Understanding the coordinate plane is crucial as it serves as a foundation for plotting and interpreting geometrical shapes and functions visually.

• The x-axis is the horizontal line, where \(y = 0\).
• The y-axis is the vertical line, where \(x = 0\).

In the exercise, lines on the coordinate plane form a bounded shape—a triangle. Vertices are the key points where lines intersect, and these are specifically critical when calculating areas.