Question

The equation $a{x}^2+bx+c=0$, where a,b,c are the sides of a $△ABC$, and the equation $x^2+\sqrt{2}x+1=0$ have a common root, the measure of $\angle C$ is

• A. 90°
• B. 45°
• C. 60°
• D. None of these

Answer 1

The correct option is D

None of these

Clearly, the roots of $x^2+\sqrt{2}x+1=0$ have imaginary roots, so one root common implies both roots are common
So, $\frac{a}{1}=\frac{b}{\sqrt{2}}=\frac{c}{1}=k$
$\therefore cosC=\frac{a^2+b^2-c^2}{2ab}=\frac{k^2+2k^2-k^2}{2k\sqrt{2}k}=\frac{1}{\sqrt{2}}$
$\Rightarrow \angle C={45}^o$