Two circles are drawn through the points (a, 5a) and (4a, a) to touch the Y-axis. Then they intersect
at an angle  ${\tan }^{-1}\left(\frac{x}{y}\right)$ then $\sqrt{x+y}=$ ………………. (x,y are co-prime)

The coordinates of the centres of these circles are $C_1(\frac{5a}{2},3a)$  and $C_2(\frac{205a}{18},\frac{29a}{3})$  respectively.
Suppose circles (v) and (vi) entersect at P and let  $\theta$  be the angle of intersection at P. Then,
   
$\cos \theta =\frac{C_1{P}^2+C_2{P}^2-C_1{C}_2^2}{2C_{1}P.C_{2}P}$ 
Now,  $C_{1}P=\frac{5a}{2},C_{2}P=\frac{205a}{18}$
And  $C_1{C}_2^2={(\frac{205a}{18}-\frac{5a}{2})}^2+{(\frac{29a}{3}-3a)}^2$
$={\left(\frac{160a}{18}\right)}^2+{\left(\frac{20a}{3}\right)}^2=\frac{40000a^2}{324}$ 
 $\therefore \cos \theta =\frac{\frac{25a^2}{2}+\frac{42025a^2}{324}-\frac{40000a^2}{324}}{2\times \frac{5a}{2}\times \frac{205a}{18}}=\frac{9}{41}$
 $\Rightarrow \tan \theta =\frac{40}{9}$
 $\Rightarrow \theta ={\tan }^{-1}\left(\frac{40}{9}\right)$