Question

Find the sum of value of x which satisfy ${\sin }^{-1}\frac{14}{\left|x\right|}+{\sin }^{-1}\frac{2\sqrt{15}}{\left|x\right|}=\frac{\pi }{2}$

• A. $-2$
• B. $0$
• C. $2$
• D. $4$

Answer 1

The correct option is B $0$

${\sin }^{-1}\frac{14}{\left|x\right|}+{\sin }^{-1}\frac{2\sqrt{15}}{\left|x\right|}=\frac{\pi }{2}$

$\Rightarrow$ ${\sin }^{-1}\frac{14}{\left|x\right|}=\frac{\pi }{2}-{\sin }^{-1}\frac{2\sqrt{15}}{\left|x\right|}={\cos }^{-1}\frac{2\sqrt{15}}{\left|x\right|}={\sin }^{-1}\sqrt{1-{\left(\frac{2\sqrt{15}}{\left|x\right|}\right)}^2}$

For $0\le \frac{2\sqrt{15}}{\left|x\right|}\le 1$ or $\left|x\right|\ge 2\sqrt{15}$, we have ${\left(\frac{14}{\left|x\right|}\right)}^2=1-{\left(\frac{2\sqrt{15}}{\left|x\right|}\right)}^2$ $\Rightarrow$ $\left|x\right|=16$

$x=\pm 16$, which satisfies $\left|x\right|\ge 2\sqrt{5}$