Question

The value of $\alpha$ for which the equation$(\mathrm{\alpha }+5){\mathrm{x}}^2-(2\mathrm{\alpha }+1)\mathrm{x}+(\mathrm{\alpha }-1)=0$ has roots equal in magnitude but opposite in sign is

• A. $\frac{7}{4}$
• B. $1$
• C. $-\frac{1}{2}$
• D. $-5$

Answer 1

The correct option is C

$-\frac{1}{2}$

Explantion for the correct option:

Find the value of $\alpha$:

$(\mathrm{\alpha }+5){\mathrm{x}}^2–(2\mathrm{\alpha }+1)\mathrm{x}+(\mathrm{\alpha }-1)=0$ roots equal in magnitude but opposite in sign.

Let $\mathrm{p}$ be a root. Then another root will be$\mathit{-}p$.

Sum of roots$=\frac{-b}{a}$

$$\begin{gathered} \Rightarrow \mathrm{p}-\mathrm{p}=\frac{(2\mathrm{\alpha }+1)}{(\mathrm{\alpha }+5)} \\ \Rightarrow 0=\frac{(2\mathrm{\alpha }+1)}{(\mathrm{\alpha }+5)} \\ \Rightarrow 2\mathrm{\alpha }+1=0 \\ \Rightarrow 2\mathrm{\alpha }=-1 \\ \Rightarrow \mathrm{\alpha }=\frac{-1}{2} \end{gathered}$$

Hence, the correct option is $C$