Question

find the value of c so that (2,0) (0,1) (4,5) (0,c) are concyclic

Answer 1

 the value of c so that (2,0) (0,1) (4,5) (0,c) are concyclic is 1 & 27/6
as-
the eqn. of circle passing through the point (2,0)
             4+4g+c²=0      ---------------------------(1)
the eqn.of circle passing through the point (0,1)
             1+2f+c² =0      -----------------------------(2)
the eqn.of circle passing through the point(4,5)
             41+8g+10f+c² =0    ------------------------(3)
from (1)                
         g= (-c²-4)÷4
from(2)  
         h=(-c²-1)÷2
putting value of g &c in (3)
             c²=27÷6
             g= (-51)/24
             f=(-33)/12
from eqn.of circle
           x²+y²-51/12x-33/6y+27/6=o
         after putting x=0  y=c
then          c=1 &  27/6