Question

Let \(α, β\) be the roots of the equation \(x^2-\sqrt{2}x+\sqrt{6}=0\) and\( \frac{1}{α^2}+1\),\(\frac {1}{β^2}+1\) be the roots of the equation \(x^2 + ax + b = 0\) . Then the roots of the equation \(x^2 – (a + b – 2)x + (a + b + 2) = 0\) are

• A. Non-real complex number
• B. Real and both negative
• C. Real and both negative
• D. Real and exactly one of them is positive

Answer 1

The Correct Option is B

\(a=\frac{-1}{\alpha^{2}}-\frac{1}{\beta^{2}}-2 \)

\(b=\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}+1+\frac{1}{\alpha^{2}\beta^{2}} \)

\(a+b=\frac{1}{(\alpha\beta)^{2}}-1=\frac{1}{6}-1=-\frac{5}{6} \)

\(x^{2}-(-\frac{5}{6}-2)x+(2-\frac{5}{6})=0 \)

\(6x^{2}+17x+7=0 \)

\(x=-\frac{7}{3} ,  x=-\frac{1}{2} \) are the roots

Both roots are real and negative.