Question

If $a^4+b^4+c^4=2c^2(a^2+b^2)$then acute value of $\angle C$is equal to

• A. ${30}^0$
• B. ${60}^0$
• C. ${45}^0$
• D. ${75}^0$

Answer 1

The correct option is C

${45}^0$

Explanation for the correct option:

Given$a^4+b^4+c^4=2c^2(a^2+b^2)$

$\Rightarrow a^4+b^4+c^4–2c^2{a}^2–2c^2{b}^2=0$

Adding $2a^2{b}^2$on both sides, we get

$\Rightarrow a^4+b^4+c^4–2c^2{a}^2–2c^2{b}^2+2a^2{b}^2=2a^2{b}^2$

$\Rightarrow {(a^2+b^2–c^2)}^2=2(ab{)}^2$

Taking square root

$(a^2+b^2–c^2)=\pm \sqrt{2}ab$

Dividing $2ab$on both sides

$\Rightarrow \frac{(a^2+b^2–c^2)}{2ab}=\frac{\pm \sqrt{2}ab}{2ab}$

$\Rightarrow \cos C=\pm \frac{1}{\sqrt{2}}$

$\Rightarrow C={45}^0$ or ${135}^0$

Acute value of angle C is ${45}^0$

Hence, option $\left(C\right)$ is the answer.