Question

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

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

Answer 1

The correct option is A $(a^2-b)x^2-2ax+1=0$

We know that, for a quadratic equation with rational coefficients, irrational roots always occur in conjugate pairs.
So, the roots of the equation are $\frac{1}{a+\sqrt{b}},\frac{1}{a-\sqrt{b}}$
Sum of the roots $=\frac{2a}{a^2-b}$
and product of the roots $=\frac{1}{a^2-b}$
Hence, the quadratic equation is
$x^2-\left(\frac{2a}{a^2-b}\right)x+\frac{1}{a^2-b}=0\Rightarrow (a^2-b)x^2-2ax+1=0$