Explanation:
(D) : Given,
The coordinate of point from which perpendicular drawn on the line is \((1, \pi)\)
We know that,
The polar coordinate representation is \((\mathrm{r}, \theta)\)
\(\therefore \mathrm{r}=1, \theta=\pi\)
\(x=r \cos \theta, y=r \sin \theta\)
The coordinate \((1,0),(1, \pi)\) and \((1, \pi / 2)\) should
\((1, \pi)=(\cos \pi, \sin \pi)\)
\(=(-1,0)\)
Similarly, \((1, \pi / 2)=(\cos \pi / 2, \sin \pi / 2)\)
\(=(0,1) \tag{i}\)
and \(\quad(1,0)=(\cos 0, \sin 0)\)
\(=(1,0) \tag{ii}\)
Now, The equation of line joining \((0,1)\) and \((1,0)\)
\(\frac{y}{x-1}=\frac{y-1}{x}\)
\(\therefore \quad x y=x y-(x+y)+1\)
\(x+y=1\)
Then, the perpendicular distance form point \((-1,0)\) to \(\mathrm{x}\) \(+\mathrm{y}=1\) is
\(\mathrm{d}=\left|\frac{-1+0-1}{\sqrt{2}}\right|\)
\(\therefore \quad \mathrm{d}=\left|\frac{-2}{\sqrt{2}}\right|\)
\(\mathrm{d}=\sqrt{2}\)