Question

If $a$ and $b$ are rational and $b$ is not a perfect square, then the quadratic equation with rational coefficient whose one root is $\frac{1}{a+\sqrt{b}}$ is

• A. $x^2-2 a x+\left(a^2-b\right)=0$
• B. $\left(a^2-\right.$ b) $x^2-2 a x+1=0$
• C. $\left(a^2-b^2\right) x^2-2 b x+1=0$
• D. $x^2+\left(a^2-b^2\right) x+\left(a^2+b^2\right)=0$

Answer 1

Answer: (B)

Clearly other root is $\frac{1}{a-\sqrt{b}}$
Hence sum of roots $=\frac{1}{a+\sqrt{b}}+\frac{1}{a-\sqrt{b}}=\frac{2 a}{a^2-b}$
and product of roots $=\frac{1}{a^2-b}$
so required quadratic equation is $x^2-\frac{2 a}{a^2-b} x+\frac{1}{a^2-b}=0$ $\Rightarrow\left(a^2-b\right) x^2-2 a x+1=0$