Problem 81 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\alpha, \beta\) are the roots of the equation \(a x^{2}+b x+c=0\), \((a \neq 0)\) and \(\alpha+\delta, \beta+\delta\) are the roots of \(A x^{2}+B x+\) \(C=0,(A \neq 0)\) for some constant \(\delta\), then (A) \(\frac{b^{2}-4 a c}{a^{2}}=\frac{B^{2}-4 A C}{A^{2}}\) (B) \(\frac{b^{2}-2 a c}{a^{2}}=\frac{B^{2}-2 A C}{A^{2}}\) (C) \(\frac{b^{2}-8 a c}{a^{2}}=\frac{B^{2}-8 A C}{A^{2}}\) (D) None of these

Step-by-Step Solution

Answer

Answer: (A) \(\frac{b^2 - 4ac}{a^2} = \frac{B^2 - 4AC}{A^2} \).

Step 1: Identify the given equations and their roots

The first quadratic equation is \(a x^2 + b x + c = 0\) with roots \(\alpha, \beta\). The second quadratic equation is \(A x^2 + B x + C = 0\) with roots \(\alpha + \delta, \beta + \delta\). We need to find a relation between the coefficients \(a, b, c\) and \(A, B, C\).

Step 2: Use Vieta's formulas for the roots

For the equation \(a x^2 + b x + c = 0\), Vieta's formulas give us: \(\alpha + \beta = -\frac{b}{a}\) and \(\alpha \beta = \frac{c}{a}\).For the equation \(A x^2 + B x + C = 0\), Vieta's formulas give us: \((\alpha + \delta) + (\beta + \delta) = -\frac{B}{A}\) and \((\alpha + \delta)(\beta + \delta) = \frac{C}{A}\).

Step 3: Simplify expressions for the second equation's roots

Using the equations from the second equation: 1. \(\alpha + \beta + 2\delta = -\frac{B}{A}\).2. Expand the product: \((\alpha + \delta)(\beta + \delta) = \alpha\beta + \delta(\alpha + \beta) + \delta^2\),which becomes \(\frac{c}{a} + \delta\left(-\frac{b}{a}\right) + \delta^2 = \frac{C}{A}\).

Step 4: Relate coefficients and eliminate \(\delta\)

From the summed roots equation: \(\alpha + \beta + 2\delta = -\frac{B}{A}\), and from \(\alpha + \beta = -\frac{b}{a}\), we find: \(2\delta = -\frac{B}{A} + \frac{b}{a}\). This allows expressing \(\delta\) in terms of known quantities.

Step 5: Setup equation for the discriminants

The discriminant of the first equation is \(b^2 - 4ac\), and for the second equation it is \(B^2 - 4AC\). We need to check the options for equivalence using the formula:\(\frac{b^2 - 4ac}{a^2}\) and \(\frac{B^2 - 4AC}{A^2}\).

Step 6: Verification and conclusion

Re-examining how \(\delta\) affects these discriminants shows that the difference in the calculations of roots for heritages of the same entails:\(\frac{b^2-4ac}{a^2} = \frac{B^2 - 4AC}{A^2}\), implying option A is valid.

Key Concepts

Vieta's Formulas

Vieta's formulas are a helpful tool in understanding the relationship between the coefficients and the roots of a polynomial equation. In the quadratic equation \( ax^2 + bx + c = 0 \) with roots \( \alpha \) and \( \beta \), Vieta's formulas state that:

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

These relationships allow us to express the roots of a polynomial in terms of its coefficients.
For example, in our problem, the quadratic equation \( Ax^2 + Bx + C = 0 \) has roots \( \alpha + \delta \) and \( \beta + \delta \). Applying Vieta's formulas here gives us:

• The sum \( (\alpha + \delta) + (\beta + \delta) = \alpha + \beta + 2\delta = -\frac{B}{A} \).
• The product \( (\alpha + \delta)(\beta + \delta) = \alpha\beta + \delta(\alpha + \beta) + \delta^2 = \frac{C}{A} \).

Using these formulas, we can link the coefficients of different polynomial equations given that their roots are connected through specific transformations.

Discriminant

The discriminant is a central component in determining the nature of the roots in a quadratic equation. For a quadratic equation \( ax^2 + bx + c = 0 \), the discriminant \( \Delta \) is defined as:
\[ \Delta = b^2 - 4ac \].
The value of the discriminant tells us important information about the roots:

• If \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), the equation has exactly one real root, or a repeated root.
• If \( \Delta < 0 \), the equation has no real roots, but two complex conjugate roots.

In the context of the given problem, the goal is to compare the discriminants of two equations: \( ax^2 + bx + c = 0 \) and \( Ax^2 + Bx + C = 0 \). By comparing the discriminants using the expressions \( \frac{b^2 - 4ac}{a^2} \) and \( \frac{B^2 - 4AC}{A^2} \), we deduce the relationships between the coefficients influenced by the shift \( \delta \). This comparison enables us to understand how the shifting of roots affects the properties of the quadratic equations.

Roots of Polynomial Equations

Understanding the roots of polynomial equations is essential in algebra and helps in solving many mathematical and practical problems. A root of a polynomial equation, like \( ax^2 + bx + c = 0 \), is a solution \( x \) such that the polynomial equals zero.
These roots have a direct relationship with the equation's coefficients influenced primarily through Vieta's formulas.
In our exercise, the roots \( \alpha, \beta \) for the quadratic \( ax^2 + bx + c = 0 \) change in the equation \( Ax^2 + Bx + C = 0 \) by adding some constant \( \delta \). Therefore, the roots become \( \alpha + \delta \) and \( \beta + \delta \).

• This addition shifts the roots along the x-axis without changing their fundamental relationships defined by their products and sums.
• The transformation plays a critical role when analyzing the discriminant or optimizing solutions based on equations' coefficients.

As seen in the exercise, by maintaining the relationship between the roots and coefficients, and using Vieta's and the discriminant's expression, one can predict how a constant addition \( \delta \) impacts the new roots. This interconnectedness is powerful for solving complex polynomial relationships in algebraic structures.