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

Question

If \(\alpha\) and \(\beta(\alpha<\beta)\) be two different real roots of the equation \(a x^{2}+b x+c=0\), then (A) \(\alpha>-\frac{b}{2 a}\) (B) \(\beta<-\frac{b}{2 a}\) (C) \(\alpha<-\frac{b}{2 a}<\beta\) (D) \(\beta<-\frac{b}{2 a}<\alpha\)

Step-by-Step Solution

Answer

C: \(\alpha < -\frac{b}{2a} < \beta\) is correct.

Step 1: Identify and Understand the Problem

We need to determine the relationship between the roots \(\alpha\) and \(\beta\) of the quadratic equation \(ax^2 + bx + c = 0\) and the vertex, \(-\frac{b}{2a}\), of the parabola represented by this quadratic function.

Step 2: Recognize the Nature of Quadratic Roots

The roots \(\alpha\) and \(\beta\) are the solutions to \(ax^2 + bx + c = 0\) and they occur where the graph of the equation, a parabola, intersects the x-axis. The vertex of this parabola is at \(-\frac{b}{2a}\), which is also the axis of symmetry.

Step 3: Compare the Vertex to the Roots

Since \(\alpha < \beta\), and for the parabola, the vertex \(-\frac{b}{2a}\) lies exactly between the roots when the parabola opens upwards (\(a > 0\)) or downwards (\(a < 0\)). Thus, \(\alpha < -\frac{b}{2a} < \beta\).

Step 4: Evaluate the Given Options

Of the given options, option (C) \(\alpha < -\frac{b}{2a} < \beta\) is the correct relationship between \(\alpha, \beta,\) and \(-\frac{b}{2a}\). This signifies that the vertex is between the two different real roots.

Key Concepts

Roots of Quadratic Equation

A quadratic equation is typically written as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. The solutions to this equation—where the parabola intersects the x-axis—are known as the roots. These roots can be real or complex numbers. Real roots occur where the graph of the equation touches or crosses the x-axis. If the quadratic has two distinct real roots, they are denoted as \( \alpha \) and \( \beta \).To find the roots, you can use the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here, \( b^2 - 4ac \) is known as the discriminant. Depending on its value:

• If the discriminant is positive, the equation has two distinct real roots.
• If it's zero, the roots are real and equal (the parabola touches the x-axis).
• If negative, the roots are complex and not real (the parabola does not intersect the x-axis).

Understanding the nature of the roots helps in visualizing how the parabola behaves.

Vertex of Parabola

The vertex of a parabola is a significant point that acts as the turning point of the graph. For the quadratic equation \( ax^2 + bx + c = 0 \), the vertex formula is given by:\[ \left( -\frac{b}{2a}, \text{value of } y \right) \]The x-coordinate of the vertex, \( -\frac{b}{2a} \), not only gives us the point at which the parabola changes direction but also provides insight into the minimum or maximum value of the parabola:

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

By understanding the vertex's position, you can easily sketch the general shape of the parabola and make predictions about the graph's behavior.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola. For the quadratic equation \( ax^2 + bx + c = 0 \), this line serves as a mirror line, dividing the parabola into two symmetrical halves.The equation for the axis of symmetry is:\[ x = -\frac{b}{2a} \]This line is critical because it helps to identify the parabola's balance point.

• For data analysis or graph sketching, the axis ensures accuracy by confirming that each point on one side has a corresponding point on the opposite side.
• Understanding this concept aids in predicting how changes to the coefficients \( a \), \( b \), and \( c \) affect the shape and position of the parabola.

Grasping the axis of symmetry enhances the ability to visualize quadratic equations' graphs, making this concept essential for anyone studying conic sections.