Question

If \(\alpha\) and \(\beta\) are the roots of \(x^{2}-3 x+a=0\) and \(\gamma, \delta\) that of \(x^{2}-12 x+b=0\) and \(\alpha, \beta, \gamma, \delta\) (in that order) form an increasing GP, then (a) \(a=2, b=32\) (b) \(a=3, b=12\) (c) \(a=4, b=16\) (d) \(\mathrm{a}=12, \mathrm{~b}=3\)

Short answer

Answer: The values of constants a and b are 2 and 32, respectively.

Step-by-step solution

Write the equations and roots relationships

The first quadratic equation given is \(x^2 - 3x + a = 0\). Let's denote its roots by \(\alpha\) and \(\beta\). Using Vieta's formulas, we have: Sum of roots: \(\alpha + \beta = 3\) Product of roots: \(\alpha \beta = a\) The second quadratic equation given is \(x^2 - 12x + b = 0\). Let's denote its roots by \(\gamma\) and \(\delta\). Using Vieta's formulas, we have: Sum of roots: \(\gamma + \delta = 12\) Product of roots: \(\gamma \delta = b\) It is also given that the roots are in increasing geometric progression: \(\alpha < \beta < \gamma < \delta\)

Use properties of geometric progression

As the roots form a geometric progression, let's denote the common ratio as 'r'. So, we can write \(\beta = \alpha r\), \(\gamma = \alpha r^2\), and \(\delta = \alpha r^3\). This can be used to derive the relationship between a and r.

Find relationship between a and r

Using the relationship from step 2, we can find the relationship between a and r. \(\alpha \beta = a\) \(\alpha \cdot (\alpha r) = a\) So we have, \(r=\frac{a}{\alpha ^{ 2 }}\)

Find \(\gamma\) and \(\delta\) in terms of \(\alpha\) and \(r\)

Using the relationships found in steps 3, we get: \(\gamma = \alpha r^2 = a\) \(\delta = \alpha r^3 = a\frac{a}{\alpha ^{ 2 }} = \frac{a^{2}}{\alpha}\)

Use the sum and product of roots for second quadratic equation

For the second quadratic equation, we have: Sum of roots: \(\gamma + \delta = a + \frac{a^{2}}{\alpha} = 12\) Product of roots: \(\gamma \delta = a \cdot \frac{a^2}{\alpha} = b\)

Solve for 'a', and find the respective 'b' values

In the options given, we have different values of a. We can plug each value of a in the equation: \(a + \frac{a^{2}}{\alpha} = 12\) and obtain corresponding b from the equation: \(b = a \cdot \frac{a^2}{\alpha}\) Checking each option: (a) if \(a=2\), then \(\alpha \beta = 2\) and \(\alpha + \beta = 3\). Solving this, we get \(\alpha = 1, \beta = 2\). Then \(\gamma=4\) and \(\delta=8\), so \(b=32\). (b) if \(a=3\), we have \(\alpha = \frac{3}{2}\), \(\beta=2\). In this case, the roots do not follow an increasing order, and it is not a valid option. (c) if \(a=4\), then \(\alpha=1\), \(\beta=4\). This does not satisfy the increasing order of roots condition, so we discard it. (d) This option also does not satisfy the condition of increasing roots, and we discard it. So the correct option is: (a) \(a=2, b=32\)

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 number, called the common ratio. For example, in a GP with the first term as 2 and a common ratio of 3, the sequence would be 2, 6, 18, 54, and so on.

In the context of quadratic equations, the roots can form a GP. For instance, if the quadratic equation has roots \( \alpha \) and \( \beta \) and \( \alpha < \beta \), and together with roots \( \gamma \) and \( \delta \) from another equation also form a GP, we can express the subsequences as \( \beta = \alpha r \) and \( \gamma = \alpha r^2 \), where 'r' is the common ratio. Understanding GPs is crucial because it allows us to relate the roots of different quadratic equations in a predictable pattern, simplifying the process of finding specific values that satisfy certain conditions.

Vieta's Formulas

Named after the mathematician François Viète (Vieta), Vieta's formulas are an elegant way of summarizing the relationship between the roots of a polynomial and its coefficients. In the case of a quadratic equation \( ax^2 + bx + c = 0 \), Vieta's formulas tell us that the sum of the roots \( \alpha \) and \( \beta \) is \( -b/a \) and the product is \( c/a \).

These formulas help us to directly determine the sum and product of the roots from the coefficients without necessarily solving the equation. By exploiting this characteristic, one can solve complex problems involving the roots of quadratic equations, like establishing the relationship between the coefficients and roots in a geometric progression.

Sum and Product of Roots

In quadratic equations, the sum and product of roots are two important properties that often simplify problem-solving. For any quadratic equation \( ax^2 + bx + c = 0 \), the sum of its roots, \( \alpha + \beta \), equals \( -b/a \), and the product, \( \alpha \beta \), equals \( c/a \).

These properties are instrumental in solving many mathematical problems because they provide a quick way to relate the coefficients of the equation to the roots. When given an equation, one can leverage these properties to guess and check solutions more efficiently, which becomes helpful in dealing with questions involving multiple quadratic equations or additional conditions, like the roots forming a geometric sequence.

Mathematical Problem-Solving

Effective mathematical problem-solving skills are fundamental for working through complex mathematical challenges, as we observe in solving quadratic equations with special conditions. A systematic approach usually involves understanding the problem, devising a plan, carrying out that plan, and revisiting the steps to check for accuracy.

Using the properties of geometric progression and Vieta's formulas to solve for the roots that fit certain criteria exemplifies a strategic approach to problem-solving. Simplifying assumptions such as a common ratio or a known sum/product of roots can lead to the application of algebraic manipulations to find the solution. This analytical technique is an essential competency in mathematics that aids students not only in homework tasks but also in their broader academic pursuits.