Problem 62 - A Complete Resource Book in Mathematics for JEE Main

Question

If the ratio of the roots of \(x^{2}+b x+c=0\) and \(x^{2}+q x+\) \(r=0\) be the same, then (A) \(r^{2} c=b^{2} q\) (B) \(r^{2} b=c^{2} q\) (C) \(r b^{2}=c q^{2}\) (D) \(r^{2}=b q^{2}\)

Step-by-Step Solution

Answer

(C) \(r b^2 = c q^2\)

Step 1: Understand the problem statement

We are given two quadratic equations: \(x^2 + bx + c = 0\) and \(x^2 + qx + r = 0\). We are told the ratio of their roots is the same. Our task is to find which relationship among the coefficients \(b, c, q,\) and \(r\) holds based on that condition.

Step 2: Expression of ratio of roots

For a quadratic equation \(ax^2 + bx + c = 0\), the roots are given by \(\alpha + \beta = -b/a\) and \(\alpha \cdot \beta = c/a\). The ratio of the roots \(\frac{\alpha}{\beta}\) can be expressed as \(\left(-\frac{b}{a} \right)^2/\left(\frac{c}{a}\right) = \frac{b^2}{c}\).

Step 3: Apply the same ratio condition

Given that the ratio is the same for both quadratics, we equate their root ratio expressions:\[\frac{b^2}{c} = \frac{q^2}{r}\]Cross-multiply to adjust the expression:\[r \cdot b^2 = c \cdot q^2\]

Step 4: Match with provided options

Now compare this result (\(r b^2 = c q^2\)) with the given multiple-choice options:(A) \(r^2 c = b^2 q\) (B) \(r^2 b = c^2 q\)(C) \(r b^2 = c q^2\)(D) \(r^2 = b q^2\)The matching option is (C).

Key Concepts

Roots of Quadratic Equations

When discussing quadratic equations, a key point to understand is their roots, which are the solutions to the equation. A quadratic equation is typically represented as \( ax^2 + bx + c = 0 \). Here, \( a, b, \) and \( c \) are coefficients, and the values of \( x \) that satisfy the equation are known as its roots. The quadratic formula, \( x = \frac{{-b \pm \sqrt{{b^2-4ac}}}}{2a} \), is used to calculate these roots. This formula gives the two possible roots \( \alpha \) and \( \beta \).
Understanding the sum and product of roots can also be quite useful. The sum of the roots \( \alpha + \beta \) is equal to \( -\frac{b}{a} \) and their product \( \alpha \beta \) is \( \frac{c}{a} \). Recognizing these relationships can provide helpful insights into the behavior of quadratic equations.

Ratio of Roots

The ratio of the roots of a quadratic equation is another intriguing and useful feature. Given the roots \( \alpha \) and \( \beta \), the ratio \( \frac{\alpha}{\beta} \) can be investigated using the properties of the sum and product of roots, derived from Vieta’s formulas.
This ratio can be expressed as:

• \( \frac{\alpha}{\beta} = \frac{\alpha}{\beta} = \frac{\left(-\frac{b}{a}\right) + \sqrt{\left(\frac{b^2}{a^2}\right) - \frac{4c}{a}}}{2c/a} \)
• It simplifies to \( \frac{b^2/c}{c/a} = \frac{b^2}{c} \)

By equating this ratio for different quadratic equations, we can find important relationships among their coefficients as we did in the exercise. This kind of comparison is valuable for solving complex problems involving multiple quadratic equations.

Coefficients of Quadratic Equations

The coefficients \( a, b, \) and \( c \) in a quadratic equation \( ax^2 + bx + c = 0 \) hold significant importance. They determine the nature and position of the parabola represented by the quadratic equation on a graph.
Let's explore the roles of these coefficients:

• \( a \): This coefficient affects the direction and width of the parabola. If \( a \) is positive, the parabola opens upwards, and if negative, it opens downwards.
• \( b \): Influences the axis of symmetry and the position of the vertex of the parabola.
• \( c \): This is the constant term and affects the position of the parabola along the vertical axis.

Understanding these coefficients allows for predicting and explaining changes in the roots and graph of a quadratic equation. In our exercise, comparing these coefficients across different equations helped us discover the relationship between them when the roots have a specific ratio.