Problem 81 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(\left(a, a^{2}\right)\) falls inside the angle made by the lines \(y=\frac{x}{2}\), \(x>0\) and \(y=3 x, x>0\), then \(a\) belongs to (A) \(\left(0, \frac{1}{2}\right)\) (B) \((3, \infty)\) (C) \(\left(\frac{1}{2}, 3\right)\) (D) \(\left(-3,-\frac{1}{2}\right)\)

Step-by-Step Solution

Answer

The correct interval is (C) \(\left(\frac{1}{2}, 3\right)\).

Step 1: Understand the Position of Points

To determine where the point \((a, a^2)\) falls, we need to understand the constraints given by the lines \(y = \frac{x}{2}\) and \(y = 3x\). For the point \((a, a^2)\) to fall inside the angle given by these lines, its coordinates must satisfy \(\frac{a^2}{a} > \frac{1}{2}\) and \(\frac{a^2}{a} < 3\). This simplifies to the inequalities \(a/2 < a < 3a\).

Step 2: Simplify the Inequalities

Since the point \((a, a^2)\) refers to a specific position, substitute \(y = a^2\) and \(x = a\). We simplify the first inequality: \(a^2 > \frac{a}{2}\) which implies \(2a^2 > a\) or \(2a^2 - a > 0\). Next, simplify the second inequality: \(a^2 < 3a\) which implies \(a(a - 3) < 0\).

Step 3: Solve the Inequality 1

Solving the inequality \(2a^2 - a > 0\), factor it as \(a(2a - 1) > 0\). This results in two intervals: either \(a > 0\) and \(2a - 1 > 0\) or \(a < 0\) and \(2a - 1 < 0\). This simplifies to \(a > \frac{1}{2}\) or \(a < 0\).

Step 4: Solve the Inequality 2

Solving the inequality \(a(a - 3) < 0\), factor it as \(a < 3\) and \(a > 0\). This results in the interval \((0, 3)\).

Step 5: Find Intersection of Intervals

Combine the results of both inequalities to find where they overlap. The intersection of the intervals \(a > \frac{1}{2}\) from inequality 1 and \((0, 3)\) from inequality 2 is \(\left(\frac{1}{2}, 3\right)\).

Step 6: Conclusion

The interval where \(a\) resides, making the point \((a, a^2)\) fall inside the angle, is option C: \(\left(\frac{1}{2}, 3\right)\).

Key Concepts

Linear Inequalities

Linear inequalities describe relationships where two expressions are not exactly equal but rather greater than or less than each other. In this problem, we deal with linear inequalities formed by the constraints of an angle between two lines, specifically the lines given by the equations \( y = \frac{x}{2} \) and \( y = 3x \).Linear inequalities define an infinite number of solutions that satisfy the conditions:

• The inequality \( a^2 > \frac{a}{2} \) means the parabola \( y = a^2 \) must lie above the line \( y = \frac{x}{2} \). This simplifies to the inequality \( 2a^2 - a > 0 \).
• The inequality \( a^2 < 3a \) indicates that the parabola must be below the line \( y = 3x \), giving us \( a(a - 3) < 0 \).

Analyzing these inequalities provides ranges of values for \( a \) that fulfill the condition of the point \( (a, a^2) \) lying within the angle. That's why solving and interpreting these inequalities is crucial in coordinate geometry.

Mathematical Reasoning

Mathematical reasoning empowers us to understand complex problems using logical steps and conclusions. To solve math problems involving coordinate geometry, like this exercise, one needs to dissect the problem into smaller, manageable parts.Here, mathematical reasoning involves:

• Understanding the position of points relative to lines on a graph, which helps in determining the relationship between \( a \) and the point \( (a, a^2) \).
• Simplifying the given conditions through algebraic manipulation to find inequalities. For instance, converting \( a^2 > \frac{a}{2} \) into \( 2a^2 - a > 0 \).
• Verifying these conditions through solving inequalities and combining their results to come to a logical conclusion, ensuring \( a \) falls within the correct interval.

Using mathematical reasoning effectively ensures clarity and precision in solving problems, especially helpful in situations involving coordinate geometry.

Algebraic Expressions

Algebraic expressions act as the cornerstone of solving coordinate geometry problems by allowing us to express complex concepts in a comprehendible form. In this exercise, understanding algebraic expressions assists in relating variables and simplifying conditions.Considerations include:

• Expressions like \( y = a^2 \), which describe the parabola's position for any given \( a \), giving us a visual understanding of where the point lies relative to other functions.
• Changing expressions into inequalities, such as from \( a^2 < 3a \) to analyzing ranges using factors, highlighting solutions within intervals.
• Combining multiple expressions and inequalities, as seen when finding overlapping solutions \( \left(\frac{1}{2}, 3\right) \), bridging the gap between abstract algebra and geometric visualization.

Grasping algebraic expressions and their manipulations is vital for anyone tackling coordinate geometry, providing the tools needed to decode and solve systematic inequalities.