Question

If \(\mathrm{p}, \mathrm{q}, \mathrm{r}\) are in \(\mathrm{GP}\) and the equations \(\mathrm{px}^{2}+2 \mathrm{qx}+\mathrm{r}=0\) and \(\mathrm{d} \mathrm{x}^{2}+2 \mathrm{ex}+\mathrm{f}=0\) have a common root, then show that \(\frac{\mathrm{d}}{\mathrm{p}}, \frac{\mathrm{e}}{\mathrm{q}}, \frac{\mathrm{f}}{\mathrm{r}}\) are in \(\mathrm{AP}\)

Short answer

Question: Show that if p, q, and r are in geometric progression and two quadratic equations \(px^2 + 2qx + r = 0\) and \(dx^2 + 2ex + f = 0\) have a common root, then the ratios of the corresponding coefficients form an arithmetic progression. Answer: When the ratios of the corresponding coefficients \(\frac{d}{p}, \frac{e}{q},\frac{f}{r}\) form an arithmetic progression, it can be shown that the given quadratic equations indeed have a common root and p, q, and r are in geometric progression.

Step-by-step solution

Set up the given equations

We are given two quadratic equations: 1. \(px^2 + 2qx + r = 0\) 2. \(dx^2 + 2ex + f = 0\) It is also given that p, q and r are in geometric progression. Let's assume that the common ratio is k, i.e., \(\frac{q}{p} = \frac{r}{q} = k\). Therefore, \(q = pk\) and \(r = qk = p{k^2}\).

Assume a common root

Let's assume that the common root is α. So, the given equations become: 1. \(p{\alpha^2} + 2q\alpha + r = 0\) 2. \(d{\alpha^2} + 2e\alpha + f = 0\) Since these two equations have a common root α, we can express the other roots of the equations as a and b respectively. So, we can write the quadratic equations as: 1. \((x - \alpha)(x - a) = px^2 + 2qx + r\) 2. \((x - \alpha)(x - b) = dx^2 + 2ex + f\)

Compare the coefficients

Now, expand the above quadratic equations and compare the coefficients: 1. \((x-\alpha)(x-a)=x^2-(\alpha+a)x+\alpha a \implies p=1, \ 2q =-(\alpha+a), \ r=\alpha a\) 2. \((x-\alpha)(x-b) = x^2-(\alpha+b)x+\alpha b \implies d=1, \ 2e=-(\alpha+b), \ f=\alpha b\)

Find the relationship between coefficients and prove the result

Now, we can find the relationship between the coefficients of the two equations: $\frac{d}{p}=1 \implies 1=1 \\ \frac{e}{q} = \frac{-(\alpha+b)}{2(-\alpha-a)} \\ \frac{f}{r} = \frac{ab}{\alpha a}$ Now, we know that q = pk and r = p{k^2}, so: $\frac{e}{q} = \frac{-(\alpha+b)}{2(-\alpha-a)k} \\ \frac{f}{r} = \frac{b}{k^2}$ Now we need to show that: \(\frac{d}{p}, \frac{e}{q},\frac{f}{r}\) are in arithmetic progression. We already know \(\frac{d}{p}=1\). Now we just need to check if \(\frac{e}{q} + \frac{f}{r} = 2 \times \frac{d}{p}\). Using the above values for the ratios: \(\frac{-(\alpha+b)}{2(-\alpha-a)k} + \frac{b}{k^2} = 2\) Multiply both sides by \(2(-\alpha-a)k^2\) to eliminate the denominators: \(-(\alpha+b)k^2 + 2b(-\alpha-a)k = 4(-\alpha-a)k^2\) Rearranging the terms: \(4(\alpha^2 + 2\alpha a + a^2)k^2 - 2(2a + b)\alpha k^2 - (\alpha+b)k^2 = 0\) Notice that \(a^2 - 2\alpha a + \alpha^2 = (\alpha - a)^2\). Substituting this back into the equation: \((\alpha - a)^2 k^2 + 2(b - 2a)\alpha k^2 + (\alpha+b)k^2 = 0\) Since \(k\) is a common factor, we can divide: \((\alpha - a)^2 k + 2(b - 2a)\alpha k + (\alpha+b)k = 0\) Now, we observe that the left side of the equation is in the form of quadratic with respect to \(k\). Thus, there must be at least one valid value of k. Therefore, the equations satisfy that the ratios of the coefficients of the quadratic equations \(\frac{d}{p}, \frac{e}{q},\frac{f}{r}\) are in arithmetic progression.

Key concepts

Geometric Progression

A Geometric Progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In mathematical terms, a GP series is represented as \(a, ar, ar^2, ar^3, \dots\), where \(a\) is the first term, and \(r\) is the common ratio.

The significance of GP in various mathematical problems, particularly in the context of IIT-JEE Mathematics, lies in its wide-ranging applications from compound interest calculations to the behavior of certain physics phenomena like wave mechanics. In the given exercise, the understanding of geometric progression is crucial as it is the basis for establishing the relationship between the coefficients of the quadratic equations.

Quadratic Equations

Quadratic equations, fundamental to algebra, are polynomial equations of degree two. The general form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). Solutions to these equations, known as roots, can be found using various methods including factoring, completing the square, or applying the quadratic formula.

The properties of the roots, such as the sum and product being related to the coefficients (by the relationships \(\alpha + \beta = -\frac{b}{a}\) and \(\alpha\beta = \frac{c}{a}\), are especially important when exploring the relationship between the coefficients and roots in complex problems, like finding a common root between two quadratic equations.

Common Root of Quadratics

The concept of common roots in quadratic equations deals with finding a value that satisfies more than one quadratic equation simultaneously. If we have two quadratics, \(ax^2 + bx + c = 0\) and \(dx^2 + ex + f = 0\), a common root \(\alpha\) would satisfy both equations. That is, \(a\alpha^2 + b\alpha + c = 0\) and \(d\alpha^2 + e\alpha + f = 0\) hold true concurrently.

Identifying common roots allows us to explore deeper relationships between the equations and is often a step towards solving more complex algebraic problems. For instance, if two quadratic equations with a common root have their coefficients in geometric and arithmetic progression, respectively, this can lead to intriguing connections, as demonstrated in the exercise.

Arithmetic Progression

An Arithmetic Progression (AP) is a sequence in which each term after the first is obtained by adding a constant, known as the common difference, to the previous term. It is denoted as \(a, a+d, a+2d, a+3d, \dots\), where \(d\) is the common difference.

Arithmetic progression is useful in various real-life scenarios, including determining time intervals or calculating the sum of evenly distributed quantities. Understanding this concept is vital when it comes to complex problems where terms of different sequences are inter-related. In our exercise, showing that the ratios \(\frac{d}{p}, \frac{e}{q},\frac{f}{r}\) form an AP from known GP relationships is essentially a test of finding harmony between these two fundamental concepts in sequences and series.