Question

The equations $x^2+b^2=1-2bx$ and $x^2+a^2=1-2ax$ have one and only one root common. Then

• A. $a-b=2$
• B. $a-b+2=0$
• C. $|a-b|=2$
• D. none of these

Answer 1

Answer: (A), (B), (C)

$a-b=2$
$a-b+2=0$
$|a-b|=2$

Given,$x^2+b^2=1-2bx$ and $x^2+a^2=1-2ax$ have only one common root.
Let $\alpha$ be the common root.
$\Rightarrow {\alpha }^2+b^2=1-2b\alpha$
${\alpha }^2+a^2=1-2a\alpha$
On substracting both the equations $b^2-a^2=\alpha (2a-2b)$
$\Rightarrow a=b$ or $\alpha =-\left(\frac{a+b}{2}\right)$
If $a=b$ they will have both roots common.
So, $\alpha =-\left(\frac{a+b}{2}\right)$Substituting $\alpha =-\left(\frac{a+b}{2}\right)$
$\Rightarrow \frac{a^2+b^2+2ab}{4}+b^2=1+b(a+b)$
$\frac{a^2+b^2+2ab}{4}=1+ab$
$a^2+b^2+2ab=4+4ab$
$\Rightarrow (a-b{)}^2=4(a-b)=\pm 2$ i.e., $(a-b)=2,a-b+2=0$
Hence, options A, B, C are correct.