Question

Given two quadratic equations $x^2-1=-b^2-2bx\&x^2-1=-a^2-2ax$ have exactly one root in common then select the correct statements.

• A. $a+b+2=0$
• B. $a=b^2+2$
• C. $a-b=0$
• D. $a=b+2$

Answer 1

The correct option is D

$a=b+2$

Given Quadratic Equations:
$x^2-1=-b^2-2bx\Rightarrow x^2+b^2+2bx=1\Rightarrow (x+b{)}^2=1\Rightarrow (x+b)=\pm 1\Rightarrow x=1-b\text{or}x=-1-b$
And similarly for other equation:
$x^2-1=-a^2-2ax\Rightarrow x^2+a^2+2ax=1\Rightarrow (x+a{)}^2=1\Rightarrow (x+a)=\pm 1\Rightarrow x=1-a\text{or}x=-1-a$
Both the equation have one common root:
$\therefore$ The possible cases are:
$A.1-b=1-a\Rightarrow a=b$
$B.1-b=-1-a\Rightarrow a-b=-2$
$C.-1-b=1-a\Rightarrow a-b=2$
$D.-1-b=-1-a\Rightarrow a=b$

Now, the condition $a=b$ is to be rejected as it will mean that both the roots are equal.
Thus $a-b=-2\&a-b=2$