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. ${90}^{\circ }$
• B. ${45}^{\circ }$
• C. ${60}^{\circ }$
• D. none of these

Answer 1

The correct option is B ${45}^{\circ }$

$x^2+\sqrt{2}x+1=0$

Roots of the above equation are imaginary.

$\therefore$ The form of the roots is $a\pm ib$.
If they have a common root then the second root is also common.
Hence the first equation is non zero multiples of others.

$\therefore \frac{a}{1}=\frac{b}{\sqrt{2}}=\frac{c}{1}$.

Hence, they form an isosceles right-angled triangle with $\angle B={90}^o$.

$\therefore \angle C={45}^o$