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If the line y=√3X+k touches the circle Xsquare+Ysquare=16, then k=??????
x2 + y2 = 16
radius of this circle = a = 4
line is touches the circle if C = a(1+m2)1/2 ............1
C is y intercept , a is radius of circle , m is slope of line
m = 31/2 , a = 4 , C = k
putting all these values in eq 1
k = 4(1+3)1/2 = 4*2 = 8
if the line touchs the circle then distance of centre of circle from the line must be the radius of circle.
and the centre of circle is (0,0).
hence distance of the line y=root 3 x +k from (0,0) is 4 units
hence-
( 0-root3 X 0-k)/root[3+1] =4
by formula of distance of a point from a line.
on solving it we get k =8
if u like my answer then approve it.
thnx.....
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