TS EAMCET 9-Sep-2020 Shift 1 Solved Paper

We have, equation of circles
S1:x2+y2−4x−6y+k=0
and S2:x2+y2+8x−4y+11=0
Let C1,C2 are centres of circles and r1,r2 are radius of circles S1 and S2 respectively.
∴‌‌C1=(2,3),C2=(−4,2)
Now, angle between circles is given by
cos‌θ=‌

r12+r22−c1c22

2r1r2

⇒‌‌‌

1

2

=‌

22−k−37

6√13−k

⇒‌‌3√13−k=−15−k
⇒‌‌9(13−k)=(15+k)2
⇒‌‌117−9k=225+k2+30k
⇒‌‌k2+39k+108=0
⇒‌‌(k+3)(k+36)=0
⇒‌‌k=−3,−36
∴‌‌k=−3