1) Circle x2+ y2 =1 cuts line x+y=k and makes the chord with length 1. No. of values of k is A) 0 B) 1 C) 2 D) 3

							
let chord of circle is AB ...drop a perpendicular on this chord it will bisect the chord , let this point be C ...
then AC = BC = 1/2
let O be the center of circle then in triangle AOC ...
AO = radius of circle , AC  = 1/2
OC  = d  ( this can be calculated by eq of line)
perpendicular distance from (0,0) to line x+y=k will be K/root2 ...
now using pythogorus theorem
AO2 = AC2 + OC2
1 = 1/4 + k2/2
 k2 = 3/2
k = + or - 3/2  so
there are 2 possible values for k
						

							
ploting the approx graph we see that the unit circle is bieng cut by the line for diff values of  k
now for k=0 the line is y=-x and its chord length is the diammeter ie 2
not moving up as we increase k to 1 we get a line through the end points of the circle ie (0,1)and(-1,0) and now the distance is
2^1/2 units which is greater than 1
clearly as we move from k=0 to k=1 the chord length is gradually decreasing
and now we still increase k
and by logic we can see that the chord length decrease and it will reach 1 for some value
as the whole thing and our arguements are symmetrical about the x axis ie for k negaive
so there are two values when this happens
so the answer is 2