Problem 92 - A Complete Resource Book in Mathematics for JEE Main

Question

If the ratio of the roots of \(a_{1} x^{2}+b_{1} x+c_{1}=0\) be equal to the ratio of the roots of \(a_{2} x^{2}+b_{2} x+c_{2}=0\), then \(\frac{a_{1}}{a_{2}}\), \(\frac{b_{1}}{b_{2}}, \frac{c_{1}}{c_{2}}\) are in (A) A.P. (B) G.P. (C) H.P. (D) None of these

Step-by-Step Solution

Answer

The ratios form a Geometric Progression (B).

Step 1: Understand the problem

We need to find the relationship between the coefficients of two quadratic equations where the ratio of their roots are equal.

Step 2: Recall the quadratic formula

The roots of a quadratic equation \(ax^2 + bx + c = 0\) are given by \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 3: Express the ratio of roots for both equations

For the equation \(a_{1}x^{2}+b_{1}x+c_{1}=0\), let the roots be \(r_1\) and \(r_2\). The ratio of the roots of the first equation is \( \frac{r_1}{r_2} = \frac{-b_{1} + \sqrt{b_{1}^2 - 4a_{1}c_{1}}}{-b_{1} - \sqrt{b_{1}^2 - 4a_{1}c_{1}}} \). Similarly, find the ratio for the second equation.

Step 4: Equate the ratios of roots

Since the ratios of the roots are given to be equal, set the expression for the first ratio equal to the expression for the second: \[ \frac{-b_{1} + \sqrt{b_{1}^2 - 4a_{1}c_{1}}}{-b_{1} - \sqrt{b_{1}^2 - 4a_{1}c_{1}}} = \frac{-b_{2} + \sqrt{b_{2}^2 - 4a_{2}c_{2}}}{-b_{2} - \sqrt{b_{2}^2 - 4a_{2}c_{2}}} \]

Step 5: Simplify the equation

Simplify each side of the equation to get a potential expression with the coefficients. The simplification will show that both expressions are equal when the ratios \( \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} \) exist.

Step 6: Identify the sequence type

When \( \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} \), it means that these ratios are constant. A constant sequence is an example of a Geometric Progression (G.P.), where every element is the same constant.

Key Concepts

Roots of Quadratic Equations

Quadratic equations are fundamental in algebra and take the form \( ax^2 + bx + c = 0 \). These equations have up to two solutions or roots, which can be real or complex numbers. Understanding the nature of these roots is essential because they reveal important properties about the graph of the equation—a parabola. The roots of a quadratic equation are determined using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, the expression \( b^2 - 4ac \) is known as the discriminant. It tells you whether the roots are real or complex:

• If the discriminant is greater than zero, the equation has two distinct real roots.
• If it is zero, the equation has exactly one real root (also known as a repeated or double root).
• If less than zero, the roots are complex numbers.

Understanding how to find and interpret these roots is crucial for solving problems involving quadratic equations.

Ratios in Algebra

Algebra often involves working with ratios, which are comparisons between two quantities. In the context of quadratic equations, you might encounter ratios when comparing coefficients or roots of similar equations. The ratio of two numbers \( a \) and \( b \) is expressed as \( \frac{a}{b} \). When applied to the roots of quadratic equations, ratios provide a meaningful way to compare different equations, especially when investigating their similarities or analyzing their behavior. In the given problem, you are comparing the ratios of the roots from two different quadratic equations. The key is to express each pair of roots and set their ratios equal, leading to useful relationships between the coefficients of the equations. When these ratios among roots are equal for two different equations, it highlights a deeper mathematical relationship that can solve the problem.

Geometric Progression

A geometric progression (G.P.) is a sequence of numbers where each term after the first is found by multiplying the previous one by a constant called the common ratio. For example, in the sequence \( 2, 4, 8, 16, \ldots \), the common ratio is 2. This pattern continues indefinitely. Geometric progressions can describe many natural phenomena where growth or decay happens at a constant rate. In the given problem, the coefficients \( \frac{a_{1}}{a_{2}}, \frac{b_{1}}{b_{2}}, \frac{c_{1}}{c_{2}} \) form a G.P. This happens because when the ratios of the roots of two quadratic equations are equal, the relationship between their coefficients also follows a consistent and uniform pattern. Understanding geometric progressions helps in solving problems like these, where relationships between multiple terms are involved. By recognizing the pattern, you can simplify complex algebraic concepts into more accessible forms.