Problem 148 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

For what values of \(a\) and \(b\) does the system \(a^{2} x-b y=a^{2}-b\) \(b x-b^{2} y=2+4 b\) possesses an infinite number of solutions?

Step-by-Step Solution

Answer

The system has infinite solutions for \(a=(-1)^{1/3}\) and \(b=(-1)^{2/3}\).

Step 1: Identify the Coefficients and Constants

In order to figure out if the lines are identical, we first identify the coefficients of \(x\) and \(y\), and the constants in both equations. In the first equation, they are \(a^{2}\), \(-b\), and \(a^{2}-b\) respectively, and in the second equation, they are \(b\), \(-b^{2}\), and \(2+4b\) respectively.

Step 2: Write down the Conditions for Identical Lines

The conditions for the system to have infinite solutions are: \n \[\frac{a^{2}}{b} = \frac{-b}{-b^{2}}\] and \[\frac{a^{2}-b}{2+4b} = \frac{a^{2}}{b}\]

Step 3: Simplify the Equations

Solving the first equation from Step 2 results in \(a^{2}=b^{3}\). Substituting this into the second equation results in \(a^{2}-a^{2/3} = 2+4a^{2/3}\). In this equation, substitute \(b=a^{2/3}\), leading us to \(a^{2}-b = 2+4b\).

Step 4: Find the solutions for \(a\) and \(b\)

We now solve \(a^{2}-b = 2+4b\) for \(a\) and \(b\). We find that \(a=(-1)^{1/3}\) and \(b=(-1)^{2/3}\).

Key Concepts

System of Equations

A system of equations refers to a set of two or more equations with the same set of variables. The goal is to find values for the variables that satisfy each equation in the system simultaneously. In our exercise, we are dealing with a system of two equations:

• Equation 1: \( a^{2} x - b y = a^{2} - b \)
• Equation 2: \( b x - b^{2} y = 2 + 4b \)

To solve a system of equations, you can use various methods, such as substitution, elimination, or graphical methods. The solution to a system of equations can be:

• a single unique solution,
• no solution, or
• infinitely many solutions.

In this exercise, we are looking for conditions under which the system has an infinite number of solutions, which often occurs when the equations are dependent, i.e., when they represent the same line in a graphical plane.

Identical Lines

For two equations to describe identical lines, every point on one line must also be a point on the other line. This means the equations do not merely intersect at one point, but coincide entirely, having the same slope and intercept.
In the context of our system of equations, having infinitely many solutions essentially means the equations define the same line:

• The first equation: \( a^{2} x - b y = a^{2} - b \)
• The second equation: \( b x - b^{2} y = 2 + 4b \)

To confirm that the lines are identical, we examine the ratios of corresponding coefficients and constants. Identical lines imply that these ratios are equal, leading to specific conditions that the coefficients and constants must satisfy.
Identifying these conditions is key to determining identical lines and consequently, infinite solutions for the system.

Coefficients and Constants

Coefficients are the numbers that multiply the variables in an equation, while constants are the standalone numbers.
In our system, the role of coefficients and constants is crucial:

• From Equation 1, the coefficients are \(a^{2}\) and \(-b\) for \(x\) and \(y\) respectively, with the constant being \(a^{2} - b\).
• From Equation 2, the coefficients are \(b\) and \(-b^{2}\), with the constant \(2 + 4b\).

Comparing the coefficients and constants between the two equations allows us to determine conditions for identical lines:

• The ratio of \(x\) coefficients: \(\frac{a^{2}}{b}\)
• The ratio of \(y\) coefficients: \(\frac{-b}{-b^{2}} = \frac{1}{b}\)
• The ratio of constants to \(x\) coefficient: \(\frac{a^{2} - b}{2 + 4b}\)

These ratios help create equations that when solved, provide the necessary conditions for infinite solutions.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations and expressions to find the desired variable values. In this exercise, it is used to determine specific values of \(a\) and \(b\) that satisfy conditions for infinite solutions.
The steps include:

• Simplifying the conditions derived from identical lines, such as \(a^{2} = b^{3}\).
• Substituting known values into other equations to simplify them further. For instance, substituting \(b = a^{2/3}\) helps solve \(a^{2} - b = 2 + 4b\).
• Finally, solving these simplified forms to find \(a\) and \(b\), resulting in solutions like \(a = (-1)^{1/3}\) and \(b = (-1)^{2/3}\).

Algebraic manipulation helps transform complex systems into simpler forms, making it easier to explore possibilities for solutions, including infinite solutions.