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

Answer: (B)

Real and both negative

Answer 2

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

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

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

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

$6x^{2}+17x+7=0$

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

Both roots are real and negative.