Question

If the roots of the equation $a{x}^2+bx+1=0$ reamians unchanged if each coefficient of the equation is increased by $1$ units, then

• A. $a\ne b$
• B. $a=b\ne 1$
• C. $a=b=1$

Answer 1

The correct option is C $a=b=1$

The equation $a{x}^2+bx+1=0$ and $(a+1)x^2+(b+1)x+2=0$ will have same roots.
$\therefore$ The ratio of coefficeints will be equal
$\Rightarrow \frac{a}{a+1}=\frac{b}{b+1}=\frac{1}{1+1}=\frac{1}{2}$
$\Rightarrow \frac{a}{a+1}=\frac{1}{2}\Rightarrow a=1$
And, similarly
$\frac{b}{b+1}=\frac{1}{2}\Rightarrow b=1$