Question

lf a,B are roots of x² - x + 2∧, =0 and a. ∦ are roots of 3x² — 10x + 27 ∧ = 0 then value of β∧/∧ :

Answer 1

Answer:

Answer:–

$\frac{ \beta \gamma }{λ} = 18$

✍ Given:–

It is given that,

$\alpha , \beta$ are the roots of the equation,

${x}^{2} - x + 2λ = 0$

And,

$\alpha ,\gamma$ are the roots of the equation,

$3 {x}^{2} - 10x + 27λ = 0$

From the first equation,

➟ $Sum of the roots,\alpha + \beta = 1$

➟ $Product of the roots, \alpha \beta = 2λ$

Again from the equation 2.

➟ $sum of the roots , \alpha + \gamma = 10/3$

➟ $product \: of \: the \: roots \: \alpha \gamma = 9λ$

So,

➟ $\frac{ \beta }{ \gamma } = \frac{2}{9}$

➟ $\beta = \frac{2 \gamma }{9}$

And also,

➟ $\beta - \gamma = 1 - \frac{10}{3}$

➟ $\beta - \gamma = \frac{ - 7}{3}$

Put the $\beta$ value in the above equation,

➟ $\frac{2 \gamma }{9} - \gamma = \frac{ - 7}{3}$

➟ $\frac{2 \gamma - 9 \gamma }{3} = - 7$

➟ $- 7 \gamma = - 21$

➟ $\gamma = 3$

So,

➟ $\beta = \frac{2 \gamma }{9}$

➟ $\beta = \frac{6}{9}$

➟ $\beta = \frac{2}{3}$

substitute $\beta$ value in the given equation,

➟ $\alpha + \beta = 1$

➟ $\alpha + \frac{2}{3} = 1$

➟ $\alpha = 1 - \frac{2}{3}$

➟ $\alpha = \frac{1}{3}$

Now,

→ $\alpha \beta = 2λ$

→ $\frac{1}{3} \times \frac{2}{3} = 2λ$

→ $9λ = 1$

→ $λ = \frac{1}{9}$

So,

The value of

➟ $\alpha = \frac{1}{3} </strong><strong>,</strong><strong> \beta = \frac{2}{3}$

➟ $\gamma = 3</strong><strong>,</strong><strong>λ</strong><strong> = \frac{1}{9}$

In the question:–

➟ $\frac{ \beta \gamma }{λ} = \frac{2}{ \frac{1}{9} }$

➟ $\frac{ \beta \gamma }{λ} = 18$

So, The answer is 18.....

Answer 2

QUESTION:

lf α, β are roots of x² - x + 2λ = 0 and α, γ are roots of 3x² - 10x + 27λ = 0 then value of βγ / λ is.

ANSWER:

• The value of βγ / λ = 18.

GIVEN:

• α, β are roots of x² - x + 2λ = 0

• α, γ are roots of 3x² - 10x + 27λ = 0

TO FIND:

• The value of βγ / λ.

EXPLANATION:

α, β are roots of x² - x + 2λ = 0.

$\boxed{ \boldsymbol{ \large{ \gray{\alpha + \beta = -\dfrac{b}{a}}}}}$

b = - 1

a = 1

$\sf \alpha + \beta = -\dfrac{ - 1}{1}$

$\sf \alpha + \beta = 1$

$\boxed{ \boldsymbol{ \large{ \gray{\alpha \ \beta = \dfrac{c}{a}}}}}$

c = 2 λ

a = 1

$\sf \alpha \ \beta = \dfrac{2 \ \lambda}{1}$

$\sf \lambda = \dfrac{ \alpha \ \beta}{2}$

α, γ are roots of 3x² - 10x + 27λ = 0

$\boxed{ \boldsymbol{ \large{ \gray{\alpha + \gamma = -\dfrac{b}{a}}}}}$

b = - 10

a = 3

$\sf\alpha + \gamma = -\dfrac{ - 10}{3}$

$\sf\alpha + \gamma = \dfrac{ 10}{3}$

$\boxed{ \boldsymbol{ \large{ \gray{\alpha \ \gamma = \dfrac{c}{a}}}}}$

c = 27 λ

a = 3

$\sf\alpha \ \gamma = \dfrac{27 \ \lambda}{3}$

$\sf\alpha \ \gamma = 9 \ \lambda$

$\sf\lambda = \dfrac{\alpha \ \gamma}{9}$

Equate both the values of λ.

$\sf \dfrac{\alpha \ \gamma}{9} = \dfrac{ \alpha \ \beta}{2}$

$\sf \dfrac{\gamma}{9} = \dfrac{\beta}{2}$

$\sf \beta = \dfrac{2}{9} \ \gamma$

$\sf We \ know \ that \ \alpha + \beta = 1$

$\sf \alpha + \dfrac{2}{9} \ \gamma = 1$

$\sf \dfrac{9 \ \alpha + 2 \ \gamma}{9} = 1$

$\sf 9 \ \alpha + 2 \ \gamma = 9$

$\sf We \ know \ that \ \alpha + \gamma = \dfrac{ 10}{3}$

$\sf \alpha = \dfrac{ 10}{3} - \gamma$

$\sf \alpha = \dfrac{ 10 - 3 \ \gamma}{3}$

$\sf 9 \left(\dfrac{ 10 - 3 \ \gamma}{3} \right) + 2 \ \gamma = 9$

$\sf 3( 10 - 3 \ \gamma) + 2 \ \gamma = 9$

$\sf 30 - 9 \ \gamma + 2 \ \gamma = 9$

$\sf - 7 \ \gamma = - 21$

$\sf \gamma = 3$

$\sf We\ know \ that\ \alpha = \dfrac{ 10}{3} - \gamma$

$\sf \alpha = \dfrac{ 10}{3} -3$

$\sf \alpha = \dfrac{ 10 - 9}{3}$

$\sf \alpha = \dfrac{1}{3}$

$\sf \dfrac{ \beta \ \gamma}{ \lambda} = \dfrac{ \beta \ \gamma}{ \dfrac{ \alpha \ \beta}{2} }$

$\sf As\ we \ know\ that\ \lambda = \dfrac{ \alpha \ \beta}{2}$

$\sf \dfrac{ \beta \ \gamma}{ \lambda} = \dfrac{2 \ \beta \ \gamma}{ \alpha \ \beta}$

$\sf \dfrac{ \beta \ \gamma}{ \lambda} = \dfrac{2 \ \gamma}{ \alpha }$

Substitute α = 1/3 and γ = 3

$\sf \dfrac{ \beta \ \gamma}{ \lambda} = \dfrac{2(3)}{ \dfrac{1}{3} }$

$\sf \dfrac{ \beta \ \gamma}{ \lambda} = 2(3) \times 3$

$\sf \dfrac{ \beta \ \gamma}{ \lambda} = 6 \times 3$

$\sf \dfrac{ \beta \ \gamma}{ \lambda} = 18$

HENCE THE VALUE OF βγ / λ = 18.