Circle

S‌=x2+y2+2kx+4y−3=0
C1‌=(−k,−2)‌ radius ‌r1=√k2+4+3
S′‌=x2+y2−4x+2ky+9=0
C2‌=(2,−k)‌ radius ‌=√4+k2−9
‌=√k2−5
Now, angle between two circles S=0 and S′=0 is
cos(180∘−θ)=‌

(r12+r22−(d2).

2r1r2

where r1,r2 are the radii and d is the distance between their centres.
‌=‌

k2+7+k2−5−(k+2)2−(k−2)2

2√k2+7√k2−5

‌⇒−cos‌θ=‌

2k2+2−(2k2+8)

2√k2+7√k2−5

‌=‌

−6

2√k2+7√k2−5

‌⇒‌‌cos‌θ=‌

3

√k2+7√k2−5

=‌

3

8

‌⇒‌‌(k2+7)(k2−9)=64
‌⇒‌‌k4+2k2−99=0
‌⇒‌‌(k2+11)(k2−9)=0⇒k2=9
‌⇒‌‌k=±3

Centre of S′=0 lies in I quadrant
[∴k=−3]
Now, radical axis of S=0 and S′=0 is
⇒‌‌‌x(2k+4)+y(4−2k)−12=0\‌−2x+10y−12=0\‌x−5y+6=0