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

Approved

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

Approved