SATHEE: Theory of Equations 3 Question 2

Theory of Equations 3 Question 2

2. A value of $b$ for which the equations $x^{2} + b x - 1 = 0$ , $x^{2} + x + b = 0$ have one root in common is

(2011)

(a) $- \sqrt{2}$

(b) $- i \sqrt{3}$

(d) $\sqrt{2}$

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Solution:

Since, $a x^{2} + b x + c = 0$ has roots $α$ and $β$ .

$\begin{aligned} \Rightarrow & α + β & = - b / a \\ and & α β & = c / a \end{aligned}$

Now, $a^{3} x^{2} + a b c x + c^{3} = 0$

On dividing the equation by $c^{2}$ , we get

$\begin{array}{ll} \frac{a^{3}}{c^{2}} x^{2} + \frac{a b c x}{c^{2}} + \frac{c^{3}}{c^{2}} = 0 \\ \Rightarrow & a \frac{a x}{c} + b \frac{a x}{c} + c = 0 \\ \Rightarrow & \frac{a x}{c} = α, β are the roots \\ \Rightarrow & x = \frac{c}{a} α, \frac{c}{a} β are the roots \end{array}$

$\begin{array}{ll} \Rightarrow & x = α β α, α β β are the roots \\ \Rightarrow & x = α^{2} β, α β^{2} are the roots \end{array}$

Divide the Eq. (i) by $a^{3}$ , we get

$\begin{aligned} x^{2} + \frac{b}{a} \cdot \frac{c}{a} \cdot x + \frac{c}{a} & = 0 \\ \Rightarrow & x^{2} - (α + β) \cdot (α β) x + (α β)^{3} & = 0 \\ \Rightarrow & x^{2} - α^{2} β x - α β^{2} x + (α β)^{3} & = 0 \\ \Rightarrow & x (x - α^{2} β) - α β^{2} (x - α^{2} β) & = 0 \\ \Rightarrow & (x - α^{2} β) (x - α β^{2}) & = 0 \end{aligned}$

$\Rightarrow x = α^{2} β, α β^{2}$ which is the required answer.

Alternate Solution

$\begin{aligned} Since, a^{3} x^{2} + a b c x + c^{3} = 0 \\ \Rightarrow x = \frac{- a b c \pm \sqrt{(a b c)^{2} - 4 \cdot a^{3} \cdot c^{3}}}{2 a^{3}} \\ \Rightarrow x = \frac{- (b / a) (c / a) \pm \sqrt{(b / a)^{2} (c / a)^{2} - 4 (c / a)^{3}}}{2} \\ \Rightarrow x = \frac{(α + β) (α β) \pm \sqrt{(α + β)^{2} (α β)^{2} - 4 (α β)^{3}}}{2} \\ \Rightarrow x = \frac{(α + β) (α β) \pm α β \sqrt{(α + β)^{2} - 4 α β}}{2} \\ \Rightarrow x = α β \frac{(α + β) \pm \sqrt{(α - β)^{2}}}{2} \\ \Rightarrow x = α β \frac{(α + β) \pm (α - β)}{2} \\ \Rightarrow x = α β \frac{α + β + α - β}{2}, \frac{α + β - α + β}{2} \\ \Rightarrow x = α β \frac{2 α}{2}, \frac{2 β}{2} \\ \Rightarrow x = α^{2} β, α β^{2} which is the required answer. \end{aligned}$