Problem 29 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(a \in[-5,30]\), then the probability that the graph of the function \(y=x^{2}+2(a+4) x-5 a+64\) is strictly above the \(x\)-axis is (A) \(\frac{27}{35}\) (B) \(\frac{8}{25}\) (C) \(\frac{8}{35}\) (D) \(\frac{17}{25}\)

Step-by-Step Solution

Answer

(C) \(\frac{8}{35}\)

Step 1: Identify the Vertex of the Parabola

The quadratic function given is \( y = x^2 + 2(a+4)x - 5a + 64 \). This is in the form \( y = ax^2 + bx + c \), where \( a=1 \), \( b=2(a+4) \), and \( c=-5a+64 \). The vertex form of a parabola is \( x = -\frac{b}{2a} \). Calculate the x-coordinate of the vertex: \( x = -\frac{2(a+4)}{2} = -(a+4) \).

Step 2: Write the Condition for the Parabola to Be Strictly Above the x-axis

The graph of the function is strictly above the x-axis if it has no real roots. The determinant condition for no real roots is \( D = b^2 - 4ac < 0 \).

Step 3: Calculate the Discriminant

The discriminant \( D = [2(a+4)]^2 - 4 \cdot 1 \cdot (-5a+64) \) simplifies to \( 4(a+4)^2 + 20a - 256 \).

Step 4: Simplify the Discriminant

Expanding \( 4(a+4)^2 \) gives \( 4(a^2 + 8a + 16) \), resulting in \( 4a^2 + 32a + 64 \). Substituting back into the discriminant gives: \( 4a^2 + 32a + 64 + 20a - 256 = 4a^2 + 52a - 192 \).

Step 5: Find Range of Values for a

We require \( 4a^2 + 52a - 192 < 0 \) to ensure that the parabola lies strictly above the x-axis. This quadratic inequality can be solved using factorization or the quadratic formula. Solving for roots by setting \( 4a^2 + 52a - 192 = 0 \) yields the critical points \( a = 3 \) and \( a = -16 \). Analyzing the inequality \( 4a^2 + 52a - 192 < 0 \) shows that \( a \) must lie between \( -16 \) and \( 3 \).

Step 6: Determine the Probability

The interval for \( a \) is given as \([-5, 30]\). The viable range for \( a \) (so that the graph is strictly above the x-axis) is \((-5, 3)\). The length of this interval is \(3 - (-5) = 8\). The total length of \([-5, 30]\) is \(30 - (-5) = 35\). Thus, the probability is \(\frac{8}{35}\).

Key Concepts

Quadratic Functions

Quadratic functions form a vital part of mathematics and can be expressed in the general form \( y = ax^2 + bx + c \). These functions graph as parabolas. The important features of quadratic functions include their shape, which is determined by the coefficient \( a \), and their vertex, which indicates the minimum or maximum point.

• If \( a > 0 \), the parabola opens upwards, and the vertex is a minimum point.
• If \( a < 0 \), the parabola opens downwards, and the vertex is a maximum point.

Quadratics are often explored for their roots—points where the graph intersects the x-axis. These roots can be found using the quadratic formula, factoring, or completing the square. Understanding the properties of quadratic functions allows us to analyze their behavior on a graph, such as determining where they lie in relation to the x-axis or how wide and steep they are.
The quadratic function presented in the exercise, \( y = x^2 + 2(a+4)x - 5a + 64 \), can be analyzed with these same principles. By examining its coefficients, we can gain insights into its shape and other attributes.

Parabola Vertex

The vertex of a parabola is a significant feature, providing us with valuable information about the function's behavior. The vertex represents the highest or lowest point on the graph, depending on the direction the parabola opens.

• The x-coordinate of the vertex is found using: \( x = -\frac{b}{2a} \).
• Once the x-coordinate is determined, substitute it back into the function to find the corresponding y-coordinate.

In the provided quadratic function, \( y = x^2 + 2(a+4)x - 5a + 64 \), we determine the vertex x-coordinate as \( x = -(a+4) \). This calculation helps us visualize where the parabola's peak or trough is located. Understanding the vertex's position is crucial for graph interpretation and analysis, especially when considering inequalities and behavior above or below the x-axis.
The vertex provides a reference point around which other calculations, like determining the range of values for \( a \), are framed.

Discriminant Analysis

Discriminant analysis is a powerful tool for understanding the nature of quadratic graphs in terms of their roots. The discriminant \( D \) is given by the formula \( D = b^2 - 4ac \). It helps us assess the behavior of the quadratic equation based on its value:

• If \( D > 0 \), there are two distinct real roots, meaning the parabola crosses the x-axis at two points.
• If \( D = 0 \), there is exactly one real root, indicating the parabola touches the x-axis at one point.
• If \( D < 0 \), there are no real roots, and the parabola does not intersect the x-axis.

In this exercise, finding that \( D < 0 \) is essential to ensure the parabola is strictly above the x-axis. For \( y = x^2 + 2(a+4)x - 5a + 64 \), through solving \( 4a^2 + 52a - 192 < 0 \), we identify conditions on \( a \) that maintain this scenario. By evaluating these inequalities, a range for \( a \) is established, which is crucial for calculating the probability that the parabola remains above the x-axis entirely.
Discriminant analysis provides a simple yet effective method for probing the possible configurations of a quadratic function and understanding its graphical representation.