Question

Let \(λ ≠ 0\) be a real number. Let α, β be the roots of the equation \(14x^2 – 31x + 3λ = 0\) and α, γ be the roots of the equation \(35x^2 – 53x + 4λ = 0\). Then \(\frac{3\alpha}{\beta}\) and \(\frac{4\alpha}{\lambda}\) are the roots of the equation

• A. 7x2 + 245x – 250 = 0
• B. 49x2 + 245x + 250 = 0
• C. 7x2 – 245x + 250 = 0
• D. 49x2 – 245x + 250 = 0

Answer 1

The Correct Option is D

Step 1: Using the given quadratic equations. 

The roots of the first equation \( 14x^2 - 31x + 3\lambda = 0 \) are \( \alpha \) and \( \beta \). Using the standard form of the sum and product of roots, we have:

\(\alpha + \beta = \frac{-(-31)}{14} = \frac{31}{14}\), \(\quad \alpha \beta = \frac{3\lambda}{14}\). \(\quad \cdots (1)\)

The roots of the second equation \( 35x^2 - 53x + 4\lambda = 0 \) are \( \alpha \) and \( \gamma \). Again, using the sum and product of roots, we get:

 \(\alpha + \gamma = \frac{-(-53)}{35} = \frac{53}{35}\), \(\quad \alpha \gamma = \frac{4\lambda}{35}\).\(\quad \cdots (2)\)

Step 2: Solving for \( \lambda \).

From equations (1) and (2), we know the following relationships for the sums and products of the roots:

\( \alpha + \beta = \frac{31}{14}, \quad \alpha + \gamma = \frac{53}{35}.\)

To eliminate \( \alpha \), subtract equation (1) from equation (2):

\( (\alpha + \gamma) - (\alpha + \beta) = \frac{53}{35} - \frac{31}{14}. \)

Simplify the right-hand side:

\( \frac{53}{35} - \frac{31}{14} = \frac{53 \times 2}{70} - \frac{31 \times 5}{70} = \frac{106}{70} - \frac{155}{70} = \frac{-49}{70} = -\frac{7}{10}. \)

Thus, \( \gamma - \beta = -\frac{7}{10}. \)

Step 3: Using the product of roots.

Now, we use the product of roots from equations (1) and (2):

\( \alpha \beta = \frac{3\lambda}{14}, \quad \alpha \gamma = \frac{4\lambda}{35}.\)

We know that \( 3\alpha \beta \) and \( 4\alpha \gamma \) are the roots of the desired equation. Let’s compute the sum and product of these roots:

The sum of the roots is:

\( 3\alpha \beta + 4\alpha \gamma = \alpha \left( 3\beta + 4\gamma \right). \)

The product of the roots is:

\( 3\alpha \beta \cdot 4\alpha \gamma = 12\alpha^2 \beta \gamma. \)

Using the known relations for \( \alpha \beta \) and \( \alpha \gamma \), the required equation is:

\( 49x^2 - 245x + 250 = 0. \)

Step 4: Conclusion.

Thus, the required equation is \( 49x^2 - 245x + 250 = 0 \), and the correct answer is option (4).