Question

The equation of the common tangent to the curves $y^2=4x$ and $x^2+32y=0$ is $x+by+c=0$ . The value of $|{\left(sin\right)}^{-1}(sin1)+{\left(sin\right)}^{-1}(sinb)+{\left(sin\right)}^{-1}\left(sinc\right)|$ is equal to

Answer 1

Answer: 1

$y=mx+\frac{1}{m}$ is tangent to $y^2=4x$
Now, $x^2+32\left(mx+\frac{1}{m}\right)=0\Rightarrow x^2+32mx+\frac{32}{m}=0$
$=>{\left(32m\right)}^2-\frac{4\times 32}{m}=0\Rightarrow m=\frac{1}{2}$
The equation of the tangent is $x-2y+4=0$
$\Rightarrow \left|{\sin }^{-1}(\sin 1)+{\sin }^{-1}(\sin (-2))+{\sin }^{-1}(\sin 4)\right|$
$=|1+2-\pi +\pi -4|=1$