Question

The perpendicular distance from the point (1, $\pi$) to the line joining (1, 0) and $(1,\frac{\pi }{2})$, (in polar coordinates) is

• A. $2$
• B. $\sqrt{3}$
• C. $1$
• D. $\sqrt{2}$

Answer 1

The correct option is D $\sqrt{2}$

$A=(1,\pi )$ comparing this with $=(r,\theta )$

$r=1$ and $\theta =\frac{\pi }{2}$ and in cartesian system point will be $(r\cos \theta ,r\sin \theta )$

so
$=(cos\pi ,sin\pi )$
$=(-1,0)$ ...(i)
similarly
$B=(1,\frac{\pi }{2})$
$=(cos\frac{\pi }{2},sin\frac{\pi }{2})$
$=(0,1)$ ...(ii)

$C=(1,0)$
$=(cos0,sin0)$
$=(1,0)$ ...(ii)

Now the equation of the line joining $B(0,1)$ and $C(1,0)$ is
$\frac{y}{x-1}=\frac{y-1}{x}$

$xy=xy-(x+y)+1$
$\to x+y=1$
The perpendicular distance of the point A from $x+y=1$ is
$d=\left|\frac{-1+0-1}{\sqrt{2}}\right|$

$=\left|\frac{-2}{\sqrt{2}}\right|$

$=|-\sqrt{2}|$
$=\sqrt{2}$