Question

If a, b and c are the sides of a triangle such that $a^4+b^4+c^4=2c^2(a^2+b^2)$ then the possible values of the angle opposite to the side c is/are?

• A. ${45}^{\circ }$
• B. ${120}^{\circ }$
• C. ${135}^{\circ }$
• D. ${150}^{\circ }$

Answer 1

Answer: (A), (C)

The correct options are
A ${45}^{\circ }$
C ${135}^{\circ }$

$a^4+b^4+c^4-2a^2{c}^2-2b^2{c}^2+2a^2{b}^2=2a^2{b}^2$
$(a^2+b^2-c^2{)}^2=(\sqrt{2}ab{)}^2$
$\Rightarrow a^2+b^2-c^2=\pm \sqrt{2}ab$
$\Rightarrow \frac{a^2+b^2-c^2}{2ab}=\pm \frac{\sqrt{2}}{2}$
$\Rightarrow cosC=\frac{1}{\sqrt{2}}$
$\Rightarrow c={45}^{\circ }$ or ${135}^{\circ }$