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:
\(\alpha + \gamma = \frac{-(-53)}{35} = \frac{53}{35}\), \(\quad \alpha \gamma = \frac{4\lambda}{35}\).\(\quad \cdots (2)\)
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, \( \gamma - \beta = -\frac{7}{10}. \)
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:
\( 49x^2 - 245x + 250 = 0. \)
Thus, the required equation is \( 49x^2 - 245x + 250 = 0 \), and the correct answer is option (4).
For real number a, b (a > b > 0), let
\(\text{{Area}} \left\{ (x, y) : x^2 + y^2 \leq a^2 \text{{ and }} \frac{x^2}{a^2} + \frac{y^2}{b^2} \geq 1 \right\} = 30\pi\)
and
\(\text{{Area}} \left\{ (x, y) : x^2 + y^2 \geq b^2 \text{{ and }} \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1 \right\} = 18\pi\)
Then the value of (a – b)2 is equal to _____.
Simplify each of the following expressions:
(i) (3 + √3)(2 + √2)
(ii) (3 + √3)(3 - √3)
(iii) (√5 + √2 )2
(iv) (√5 - √2)(√5 + √2)