Let A=(1,2), B=(3,4)  and let C=(x,y) be a point such that (x−1)(x−3)+(y−2)(y−4)=0 .If arc(ΔABC)=1 then maximum number of positions of C in the xy−  plane is

(x−1)(x−3)+(y−2)(y−4)=0 ⇒   AC⊥BC  ⇒∠ACB=900∴ C is on the circle whose diameter is ABAB=22                  As arc(ΔABC)=1,12.22.(altitude)=1∴  altitude =12< radius. So, there are four possible positions of CNote : If (ΔABC)=2, altitude=radius⇒  two possible positions of C