"Let $\lambda$ $\ne$ 0 be in R. If $\alpha$ and $\beta$ are the roots of the equation x2 $-$ x + 2$\lambda$ = 0, and $\alpha$ and $\gamma$ are the roots of equation 3x2 $-$ 10x + 27$\lambda$ = 0, then ${{\beta \gamma } \over \lambda }$ is equal to ____________."

Question

Accepted Answer

"3$\alpha$² $-$ 10$\alpha$ + 27$\lambda$ = 0 ..... (1)

$\alpha$² $-$ $\alpha$ + 2$\lambda$ = 0 ...... (2)

(1) $-$ 3(2) gives

$-$7$\alpha$ + 21$\lambda$ = 0 $\Rightarrow$ $\alpha$ = 3$\lambda$

Put $\alpha$ = 3$\lambda$ in equation (1) we get

9$\lambda$² $-$ 3$\lambda$ + 2$\lambda$ $-$ 0

9$\lambda$² = $\lambda$ $\Rightarrow$ $\lambda$ = ${1 \over 9}$ as $\lambda$ $\ne$ 0

Now, $\alpha$ = 3$\lambda$ $\Rightarrow$ $\lambda$ = ${1 \over 3}$

$\alpha$ + $\beta$ = 1 $\Rightarrow$ $\beta$ = 2/3

$\alpha$ + $\gamma$ = ${10 \over 3}$ $\Rightarrow$ $\gamma$ = 3

$${{\beta \gamma } \over \lambda } = {{{2 \over 3} \times 3} \over {{1 \over 9}}} = 18$$"

Let $\lambda$ $\ne$ 0 be in R. If $\alpha$ and $\beta$ are the roots of the equation x² $-$ x + 2$\lambda$ = 0, and $\alpha$ and $\gamma$ are the roots of equation 3x² $-$ 10x + 27$\lambda$ = 0, then ${{\beta \gamma } \over \lambda }$ is equal to ____________.

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Let $\lambda$ $\ne$ 0 be in R. If $\alpha$ and $\beta$ are the roots of the equation x2 $-$ x + 2$\lambda$ = 0, and $\alpha$ and $\gamma$ are the roots of equation 3x2 $-$ 10x + 27$\lambda$ = 0, then ${{\beta \gamma } \over \lambda }$ is equal to ____________.

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Let $\lambda$ $\ne$ 0 be in R. If $\alpha$ and $\beta$ are the roots of the equation x² $-$ x + 2$\lambda$ = 0, and $\alpha$ and $\gamma$ are the roots of equation 3x² $-$ 10x + 27$\lambda$ = 0, then ${{\beta \gamma } \over \lambda }$ is equal to ____________.