Question

Number of value $x$ satisfying the equation ${\sin }^{-1}\left(\frac{5}{x}\right)+{\sin }^{-1}\left(\frac{12}{x}\right)=\frac{\pi }{2}$ is

• A. 0
• B. 1
• C. 2
• D. more than 2

Answer 1

Answer: (B)

1

${\sin }^{-1}\left(\frac{5}{x}\right)+{\sin }^{-1}\left(\frac{12}{x}\right)=\frac{\pi }{2}$
${\sin }^{-1}\left(\frac{5}{x}\right)=\frac{\pi }{2}-{\sin }^{-1}\left(\frac{12}{x}\right)$
${\sin }^{-1}\left(\frac{5}{x}\right)={\cos }^{-1}\left(\frac{12}{x}\right)$
${\cos }^{-1}\left(\frac{\sqrt{x^2-25}}{x}\right)={\cos }^{-1}\left(\frac{12}{x}\right)$
$x^2-25=144$
$x^2=169$
$x=\pm 13$
However for -13, we wouldn't get $\frac{-\pi }{2}$ on the RHS.
Hence there is only one solution, $x=13$