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

To solve the problem, we need to understand that α, β, and γ are the roots of the given quadratic equations. Here's how we approach this:

1. Firstly, consider the equation \(14x^2 - 31x + 3λ = 0\) with roots α and β. By Vieta's formulas, we have:
   • Sum of roots: α + β = \frac{31}{14}
   • Product of roots: αβ = \frac{3λ}{14}
2. Next, consider the equation \(35x^2 - 53x + 4λ = 0\) with roots α and γ. Applying Vieta's formulas again:
   • Sum of roots: α + γ = \frac{53}{35}
   • Product of roots: αγ = \frac{4λ}{35}
3. To find the roots of the equation formed by \(\frac{3α}{β}\) and \(\frac{4α}{λ}\), we calculate:
   • Sum: \(\frac{3α}{β} + \frac{4α}{λ} = \frac{3αλ + 4αβ}{βλ} = \frac{3}{4} + 4 = \frac{53}{12}\)
   • Product: \(\frac{3α}{β} \cdot \frac{4α}{λ} = \frac{12α^2}{βλ} = \frac{12 \cdot 14}{3 \cdot 35} = \frac{168}{105} = \frac{8}{5}\)
4. The quadratic equation must match the behaviors derived from these expressions. The quadratic equation with sum \(\frac{53}{12}\) and product \(\frac{8}{5}\) is:

   x^2 - \left(\frac{53}{5}\right)x + \frac{8}{5} = 0

5. By multiplying through by a common factor to clear fractions, we express this in the form given in the options. Thus, the standardized version is:

   49x^2 - 245x + 250 = 0

After verifying with the given options, the correct option for the roots \(\frac{3α}{β}\) and \(\frac{4α}{λ}\) being solved by the equation is 49x² – 245x + 250 = 0.