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 $\left|\right.\left(sin\right)^{- 1}\left(\right.sin1\left.\right)+\left(sin\right)^{- 1}\left(\right.sinb\left.\right)+\left(sin\right)^{- 1}\left(sin c\right)\left|\right.$ is equal to

Answer 1

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