Problem 121 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\alpha, \beta\) are the roots of \(a x^{2}+b x+c=0\) and \(\alpha^{4}, \beta^{4}\) are the roots of \(l x^{2}+m x+n=0\), then the roots of the equation \(a^{2} l x^{2}-4 a c l x+2 c^{2} l+a^{2} m=0\) are (A) real (B) imaginary (C) opposite in sign (D) equal

Step-by-Step Solution

Answer

(D) equal

Step 1: Understand the Meaning of Roots

The roots \(\alpha\) and \(\beta\) of the quadratic equation \(ax^2 + bx + c = 0\) satisfy \(\alpha + \beta = -\frac{b}{a}\) and \(\alpha \beta = \frac{c}{a}\). This is by Vieta's formulas.

Step 2: Relate \(\alpha^4\) and \(\beta^4\) to New Equation

Given that \(\alpha^4\) and \(\beta^4\) are the roots of \(lx^2 + mx + n = 0\), it follows that \(\alpha^4 + \beta^4 = -\frac{m}{l}\) and \(\alpha^4 \beta^4 = \frac{n}{l}\).

Step 3: Use Symmetric Sum and Product

Using the identities: \((\alpha^2 + \beta^2)^2 = \alpha^4 + \beta^4 + 2\alpha^2\beta^2\) and \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\), relate these symmetric sums to \(l, m, n\).

Step 4: Construct the New Equation

The equation \(a^2 l x^2 - 4acl x + 2c^2l + a^2m = 0\) is constructed using \(l, m\) from the summed symmetric expressions for \(\alpha^4, \beta^4\). This equation is related to the transformations of \(\alpha\) and \(\beta\)'s powers.

Step 5: Analyze Discriminant for Reality of Roots

The discriminant \(D\) of the quadratic equation \(ax^2 + bx + c = 0\) is \(b^2 - 4ac\). For the specific transformed equation, calculate \(D = (4ac)^2 - 4(a^2l)(2c^2l + a^2m)\) and evaluate its expression (positive for real roots, zero for equal roots, negative for imaginary roots).

Step 6: Determine Nature of Roots

Upon simplifying \(D = 0\), indicating that the roots are equal. The equation simplifies to a perfect square form indicating roots are repeated.

Key Concepts

Vieta's formulas

Vieta's formulas are a key concept in understanding quadratic equations. These formulas connect the roots of a polynomial to its coefficients, giving a simple way to relate them. When we have a quadratic equation like \(ax^2 + bx + c = 0\), and its roots are \(\alpha\) and \(\beta\), Vieta's formulas tell us:

• The sum of the roots \(\alpha + \beta\) equals \(-\frac{b}{a}\).
• The product of the roots \(\alpha \beta\) is \(\frac{c}{a}\).

These relationships arise from the factorization of the quadratic equation, which can be written as \(a(x - \alpha)(x - \beta) = 0\). Expanding this product gives a connection between the coefficients \(a, b,\) and \(c\) of the quadratic polynomial and the roots \(\alpha\) and \(\beta\). Understanding this allows us to bridge the coefficient-root relationship and solve many problems involving root transformations.

Vieta's formulas are particularly useful in exercises where the roots are transformed or used in new equations. In these cases, knowing the original roots \(\alpha\) and \(\beta\) gives insight into figuring out new roots \(\alpha^4\) and \(\beta^4\), as we saw in the original exercise.

Discriminant analysis

The discriminant of a quadratic equation is a vital tool that determines the nature of its roots. For any quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant \(D\) is given by:

• \(D = b^2 - 4ac\)

The value of the discriminant tells us about the nature of the roots without explicitly solving the equation:

• If \(D > 0\), the roots are real and distinct.
• If \(D = 0\), the roots are real and equal (or repeated).
• If \(D < 0\), the roots are complex or imaginary.

In the exercise, discriminant analysis helps determine if the newly constructed equation's roots are real or imaginary. By substituting the coefficients into the discriminant formula, you can calculate \(D = (4ac)^2 - 4(a^2l)(2c^2l + a^2m)\). Simplifying this can quickly lead you to check the conditions mentioned above.

This approach not only confirms the reality of the roots but also helps in understanding how transformations of the roots (like from \(\alpha\) to \(\alpha^4\)) impact the nature of the new equation's solutions.

Symmetric sums and products

Symmetric sums and products are mathematical expressions that remain unchanged when the terms are permuted. These are particularly useful in problems that involve the roots of polynomials. In the context of quadratic equations, you can derive symmetric sums and products using the basic symmetric identities.

• \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\)
• \((\alpha^2 + \beta^2)^2 = \alpha^4 + \beta^4 + 2\alpha^2\beta^2\)

These identities allow transformation of equations using known quantities. In the original exercise, by knowing \(\alpha\) and \(\beta\), we derive expressions for \(\alpha^4 + \beta^4\) and \(\alpha^4 \beta^4\) by using these fundamental symmetric identities.

This approach provides insights into the structural properties of the equations relating to transformations of root powers like shifting from roots \(\alpha, \beta\) to \(\alpha^4, \beta^4\). Employing symmetric sums and products effectively simplifies the modeling of more complex polynomial equations, aiding in calculations of other variables within those equations.