Question

If the equations $x^2+b^2=1-2bx$ and $x^2+a^2=1-2ax$ have one and only one common root, then the value of $|a-b|$ is

• A. $0$
• B. $2$
• C. None of these
• D. $1$

Answer 1

The correct option is B $2$

Here $a\ne b$,
$\because$ For $a=b$ two equations become identical, so they have two roots in common.
Let $\alpha$ be the common root of the given equations. Then,
${\alpha }^2+b^2=1-2b\alpha \cdots \left(1\right){\alpha }^2+a^2=1-2a\alpha \cdots \left(2\right)$
Using equation $\left(1\right)$ and $\left(2\right)$,
$b^2-a^2=2(a-b)\alpha \Rightarrow \alpha =-\frac{(a+b)}{2}$
Putting the value in equation $\left(1\right)$,
$\frac{(a+b{)}^2}{4}+b^2=1+2b\times \frac{a+b}{2}\Rightarrow (a+b{)}^2+4b^2=4+4b^2+4ab\Rightarrow (a+b{)}^2-4ab=4\Rightarrow (a-b{)}^2=4\therefore |a-b|=2$