Since, angle θ between two circles is given by cos‌θ=‌
d2−r12−r22
2r1r2
, where d is the distance between the centers. Given, θ=cos−1(‌
1
4
) ⇒‌‌cos‌θ‌=‌
1
4
d2‌=(1−‌
k
2
)2+(‌
−k
2
−1)2 ‌=1−k+‌
k2
4
+1+k+‌
k2
4
‌=2+‌
k2
2
Since k<0,r1=‌
−k
2
and r2=‌
−k
2
Substituting into the formula ‌‌
1
4
=‌
(2+
k2
2
)−(
−k
2
)2−(
−k
2
)2
2(‌
−k
2
)(‌
−k
2
)
‌⇒‌‌‌
1
4
=‌
2+
k2
2
−
k2
4
−
k2
4
2(‌
k2
4
)
‌⇒‌‌‌
1
4
=‌
4
k2
‌⇒‌‌k2=16 ‌⇒‌‌k=±4 But since k<0,k=−4 Now, the equation of the radical axis is ‌S1−S2=0 ‌(x2+y2−2x+ky+1)−(x2+y2−kx. ‌−2y+1)=0 ‌⇒−2x+ky+kx+2y=0 ‌⇒−6x−2y=0‌‌(∵k=−4) ‌⇒3x+y=0 The points satisfying this equation are (−1,3) and (1,−3) lies on the radical axis.