JEE Advanced · Mathematics · 2. Quadratic Equations
The value of \(b\) for which the equations \(x^2+b x-1=0, x^2+x+b=0\) have one root in common is
- A \(-\sqrt{2}\)
- B \(-i \sqrt{3}\)
- C \(i \sqrt{5}\)
- D \(\sqrt{2}\)
Answer & Solution
Correct Answer
(B) \(-i \sqrt{3}\)
Step-by-step Solution
Detailed explanation
If \(a_1 x^2+b_1 x+c_1=0\)
and \(a_2 x^2+b_2 x+c_2=0\)
have a common real root, then
\(\Rightarrow \left(a_1 c_2-a_2 c_1\right)^2=\left(b_1 c_2-\right. \left.b_2 c_1\right) \)
\( \left(a_1 b_2-a_2 b_1\right)\)
\(\left.\therefore \begin{array}{l}x^2+b x-1=0 \\ x^2+x+b=0\end{array}\right\}\) have a common root.
\(\Rightarrow (1+b)^2=\left(b^2+1\right)(1-b)\)
\(\Rightarrow b^2+2 b+1=b^2-b^3+1-b\)
\(\Rightarrow b^3+3 b=0 \Rightarrow b\left(b^2+3\right)=0\)
\(\Rightarrow b=0, \pm \sqrt{3} i\)
and \(a_2 x^2+b_2 x+c_2=0\)
have a common real root, then
\(\Rightarrow \left(a_1 c_2-a_2 c_1\right)^2=\left(b_1 c_2-\right. \left.b_2 c_1\right) \)
\( \left(a_1 b_2-a_2 b_1\right)\)
\(\left.\therefore \begin{array}{l}x^2+b x-1=0 \\ x^2+x+b=0\end{array}\right\}\) have a common root.
\(\Rightarrow (1+b)^2=\left(b^2+1\right)(1-b)\)
\(\Rightarrow b^2+2 b+1=b^2-b^3+1-b\)
\(\Rightarrow b^3+3 b=0 \Rightarrow b\left(b^2+3\right)=0\)
\(\Rightarrow b=0, \pm \sqrt{3} i\)
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