UPSC NDA I 2016 Mathematics Solved Paper

Concept:The distance between centers of two circles is found using the center coordinates from their equations.Explanation:For the first circle $(x-1)^2+(y-3)^2=r^2$, the center is $(1,3)$.For the second circle $x^2+y^2-8x+2y+8=0$, rewrite in standard form to find center.Comparing with $x^2+y^2+2gx+2fy+c=0$, we have $2g=-8$ so $g=-4$, and $2f=2$ so $f=1$.Thus the center is $(-g,-f) = (4,-1)$.Now apply distance formula between $(1,3)$ and $(4,-1)$:$$\sqrt{(4-1)^2+(-1-3)^2}=\sqrt{3^2+(-4)^2}=\sqrt{9+16}=\sqrt{25}=5$$So the distance is $5$ units.Answer:5 units (Option A).