If alpha, beta and gamma are three consecutiv | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Since $\alpha, \beta, \gamma$ are in $G.P.$,we have $\beta^2 = \alpha\gamma$. Given that the equations $\alpha x^2 + 2\beta x + \gamma = 0$ and $x

Vedclass · Accepted Answer

$\beta\gamma$

Vedclass · Answer

(D) Since $\alpha, \beta, \gamma$ are in $G.P.$,we have $\beta^2 = \alpha\gamma$. Given that the equations $\alpha x^2 + 2\beta x + \gamma = 0$ and $x^2 + x - 1 = 0$ have a common root,and the coefficients are proportional,both roots must be common. Thus,$\frac{\alpha}{1} = \frac{2\beta}{1} = \frac{\gamma}{-1} = k$. This implies $\alpha = k$,$\beta = \frac{k}{2}$,and $\gamma = -k$. Substituting these into the expression $\alpha(\beta + \gamma)$: $\alpha(\beta + \gamma) = k(\frac{k}{2} - k) = k(-\frac{k}{2}) = -\frac{k^2}{2}$. Also,$\beta\gamma = (\frac{k}{2})(-k) = -\frac{k^2}{2}$. Therefore,$\alpha(\beta + \gamma) = \beta\gamma$.

If $\alpha, \beta$ and $\gamma$ are three consecutive terms of a non-constant $G.P.$ such that the equations $\alpha x^2 + 2\beta x + \gamma = 0$ and $x^2 + x - 1 = 0$ have a common root,then $\alpha(\beta + \gamma)$ is equal to

Similar Questions

The common roots of the equations $x^{12} - 1 = 0$ and $x^4 + x^2 + 1 = 0$ are:

If $ax^2 + bx + c = 0$ and $bx^2 + cx + a = 0$ have a common root,where $a \neq 0$,then $\frac{a^3 + b^3 + c^3}{abc} = $

Let the equations $ax^2-7x+c=0$ and $ax^2+5x-c=0$ have a common root and $ac \neq 0$. If $3$ is a root of $ax^2-7x+c=0$ other than the common root,then the common root of the given equations is

Let $\lambda \neq 0$ be in $\mathbb{R}$. If $\alpha$ and $\beta$ are the roots of the equation $x^{2}-x+2\lambda=0$,and $\alpha$ and $\gamma$ are the roots of the equation $3x^{2}-10x+27\lambda=0$,then $\frac{\beta\gamma}{\lambda}$ is equal to:

If the equations $x^2 - 11x + a = 0$ and $x^2 - 14x + 2a = 0$ have a common factor and $a \neq 0$,then what is the common factor?

Vedclass Test Series

Exam Paper Generator

Online Exam Module