Problem 7 - A Complete Resource Book in Mathematics for JEE Main

Question

The point \((1, \beta)\) lies on or inside the triangle formed by the lines \(y=x, x\)-axis and \(x+y=8\), if (A) \(0<\beta<1\) (B) \(0 \leq \beta \leq 1\) (C) \(0<\beta<8\) (D) none of these

Step-by-Step Solution

Answer

The correct option is (B) \(0 \leq \beta \leq 1\).

Step 1: Identify Triangle Boundaries

The problem involves finding the region defined by the lines \(y = x\), the \(x\)-axis (\(y = 0\)), and \(x + y = 8\). These lines form a triangle in the coordinate plane. We will identify the bounds of this triangle.

Step 2: Determine Intersection Points

Find where these lines intersect: - \(y = x\) and \(x + y = 8\) intersect at \((4,4)\).- \(y = x\) and the \(x\)-axis intersect at the origin \((0,0)\).- \(x + y = 8\) and the \(x\)-axis intersect at \((8,0)\).Thus, the triangle has vertices at \((0,0), (4,4), (8,0)\).

Step 3: Position Point on Triangle

The point \((1, \beta)\) must lie within or on the edges of this triangle. For this, \(0 \leq \beta \leq x\) (the region under \(y=x\)) and \(0 \leq \beta \leq 8-x\) (the region under \(x+y=8\)), specifically for \(x = 1\): - From \(y = x\), \(0 \leq \beta \leq 1\).- From \(x+y=8\), \(\beta \leq 7\), but this is less restrictive since the intersection is limited below \(x\).

Step 4: Find Conditions on \(\beta\)

Since the region constrained by the line \(y = x\) for \(x=1\) is the smallest effective bound, we use \(0 \leq \beta \leq 1\) to determine the valid range for \(\beta\), including endpoints to encapsulate points on the line.

Key Concepts

Triangle Properties

Understanding the properties of a triangle is fundamental in coordinate geometry. In this context, we focus on the triangle formed by lines on a coordinate plane. Specifically, these lines are given as follows: the line \(y = x\), the \(x\)-axis (which is \(y = 0\)), and the line \(x + y = 8\). To define a triangle, we need to find the points where these lines intersect:

• \(y = x\) and \(x+y = 8\) intersect at the point \((4,4)\).
• The line \(y = x\) meets the \(x\)-axis at the origin point \((0,0)\).
• Lastly, the line \(x + y = 8\) and the \(x\)-axis intersect at \((8,0)\).

These intersection points \((0,0), (4,4), (8,0)\) serve as the vertices of the triangle. Understanding these vertex points lets us visualize the triangular region on the coordinate plane. Remember, these properties help in determining inclusion of points like \((1, \beta)\) within the triangle.

Lines and Angles

In coordinate geometry, lines and angles within a triangle play a significant role in determining boundaries and intersections. For this exercise, the lines \(y = x\), \(x + y = 8\), and the \(x\)-axis create the sides of the triangle.

- The line \(y = x\) forms a 45-degree angle with the \(x\)-axis due to its slope of 1.
- The line \(x + y = 8\) can also be rewritten in the form \(y = -x + 8\), indicating a line with a negative slope, crossing both axes.

The essence of the lines and angles in triangle geometry is how they direct the region containing potential points of interest. The line equations act as barriers that define the boundaries within the triangular region. By understanding their slopes and intercepts, we can better predict potential solutions.

Inequalities in Geometry

To determine if a point like \((1, \beta)\) is inside or on the triangle, understanding inequalities is vital. These inequalities are formed from the line equations that shape the triangle and confine or allow inclusion within its space.

For instance, the line \(y = x\) suggests that any point \((x, y)\) should satisfy the inequality \(0 \leq y \leq x\) for its inclusion below this line. Similarly, the line \(x + y = 8\) implies that the point must also satisfy \(0 \leq y \leq 8 - x\).

Specifically analyzing the point \((1, \beta)\):

• The inequality for line \(y = x\) is \(0 \leq \beta \leq 1\), confining \(\beta\) to be less than or equal to \(x\), ensuring it lies under or on the line.
• For the line \(x + y = 8\), a less restrictive condition \(\beta \leq 7\) exists, ensuring the point is inside or on that boundary as well.

Thus, for \(0 \leq \beta \leq 1\), the inequalities give a complete view of the geometric constraints, aiding the determination of the point's position concerning the triangle.