Question

If \(a x^{2}+2 b x+c=0\) and \(a_{1} x^{2}+2 b_{1} x+c_{1}=0\) have a common root, and if \(\frac{a}{a_{1}}, \frac{b}{b_{1}}, \frac{c}{c_{1}}\) are in AP then \(a_{1}, b_{1}, c_{1}\) are in (a) GP (b) AP (c) HP (d) AGP

Short answer

Answer: (b) AP

Step-by-step solution

Write down the given information

The two quadratic equations given are: 1. \(ax^2 + 2bx + c = 0\) 2. \(a_1x^2 + 2b_1x + c_1 = 0\) The ratios of their coefficients are in arithmetic progression (AP): \(\frac{a}{a_1}, \frac{b}{b_1}, \frac{c}{c_1}\)

Write down the condition for the coefficients to be in AP

For the ratios of the coefficients to be in the AP, the middle term should be the average of the other two terms. Mathematically, we can write this as: \(\frac{b}{b_1} = \frac{\frac{a}{a_1} + \frac{c}{c_1}}{2}\)

Find the relationship between the coefficients

We can rewrite the above equation as: \(\frac{b}{b_1} = \frac{a}{a_1} + \frac{c}{c_1} - \frac{a}{a_1}\) Now, we can see that \(\frac{b}{b_1} - \frac{a}{a_1} = \frac{c}{c_1} - \frac{a}{a_1}\). This shows that subtracting the first coefficient's ratio from the others results in the same difference (constant difference). Hence, the coefficients of the second equation, \(a_1, b_1, c_1\), are in arithmetic progression (AP).

Choose the correct option

The relationship between the coefficients of the second equation is arithmetic progression (AP). Therefore, the answer is: (b) AP

Key concepts

Arithmetic Progression

An Arithmetic Progression (AP) is one of the fundamental concepts in algebra. It refers to a sequence of numbers where each term after the first is obtained by adding a constant, called the common difference, to the previous term. This common difference can be positive, negative, or zero, leading to increasing, decreasing, or constant sequences respectively.

Mathematically, an AP series is represented as: \(a, a + d, a + 2d, a + 3d, \ldots\), where \(a\) is the first term, and \(d\) is the common difference. The nth term of an AP can be found using the formula \(a_n = a + (n-1)d\). It's crucial for students to grasp AP, as many real-world phenomena can be modeled using this simple yet powerful concept.

Understanding AP in Quadratics

When dealing with quadratic equations and their coefficients in the form of \(ax^2+bx+c\), recognizing patterns in their progression can significantly influence the behavior of the equation's solutions. In the context of our exercise, understanding AP helps in determining the structure of the coefficients when given a common root.

Quadratic Coefficients

Quadratic coefficients are the numerical factors that multiply the terms in a quadratic equation, which is a polynomial equation of the second degree. The general form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are the coefficients, and \(x\) represents the variable or unknown. The coefficient \(a\) multiplies the squared term \(x^2\), \(b\) multiplies the linear term \(x\), and \(c\) is the constant term.

These coefficients determine the shape and position of the parabola that represents the equation on a graph. The value of \(a\) affects the parabola's opening direction and width, \(b\) affects the location of the vertex along the x-axis, and \(c\) gives the point where the parabola crosses the y-axis.

Significance in Equations with a Common Root

The coefficients play a pivotal role when two quadratic equations share a common root. The relationship between the coefficients, especially in the context of AP, provides insight into the solutions' nature and can assist in determining if the common root condition is met.

Common Root Condition

The common root condition for quadratic equations is a scenario where two different quadratic equations share at least one solution or root. This indicates a certain relation among their coefficients. Finding a common root between two different quadratics can lead to the discovery of underlying patterns and relationships that can be quite insightful.

For instance, if two quadratic equations \(ax^2 + bx + c = 0\) and \(a_1x^2 + b_1x + c_1 = 0\) have a common root, and the ratios of their corresponding coefficients are in arithmetic progression (AP), this leads to a set of conditions governing these coefficients.

As seen in the given exercise, analyzing the relationship between coefficients using the concept of AP helped us establish that \(a_1, b_1, c_1\) are also in AP. This showcases one way in which AP can define or describe the characteristics of quadratic equations. Such insights are crucial for students as they can simplify complex problems and lead to more intuitive understanding.