Question

The equation of one side of a square is $3\cos \theta +4\sin \theta =\frac{5}{r}$ and its one vertex is $(\sqrt{2},\frac{\pi }{4})$ then the length of its side is

• A. $\frac{1}{5}$
• B. $\frac{2}{5}$
• C. $\frac{3}{5}$
• D. $\frac{4}{5}$

Answer 1

The correct option is B $\frac{2}{5}$

We have $x=r\cos \theta ,y=r\sin \theta$
Given equation $3\cos \theta +4\sin \theta =\frac{5}{r}$
$\Rightarrow 3r\cos \theta +4r\sin \theta =5$
$\Rightarrow 3x+4y=5$
Given point is $(\sqrt{2},\frac{\pi }{4})$
$\Rightarrow x=1,y=1$
So, the point is $(1,1)$
Now the perpendicular distance from $(1,1)$on the line $3x+4y=5$
$p=\frac{|2|}{\sqrt{9+16}}$
$\Rightarrow p=\frac{2}{5}$
So, the length of side of square is $\frac{2}{5}.$