TS EAMCET 03 May 2019 Shift 2 Solved Paper

The equation of the circle passing through intersection of line
x+y+1=0 and circle x2+y2+x+3y=0 is
x2+y2+x+3y+λ(x+y+1)=0
x2+y2+(1+λ)x+(3+λ)y+λ=0
Let the intersection points of tangents A and B is (h,k). Then
xh+yk+‌

x+h

2

+‌

3y+3k

2

=0
(2h+1)x+(2k+3)y+h+3k=0
since x+y+1=0 is identical to the above line.
‌

2h+1

1

=‌

2k+3

1

=‌

h+3k

1

h=2‌ and ‌k=1
This implies (2,1) is the point of the circle. Then,
x2+y2+(1+λ)2+(3+λ)+λ‌‌=0
λ‌‌=−‌

5

2

The centre of the circle is
(−‌

1

2

(1−‌

5

2

),−‌

1

2

(3−‌

5

2

))=(‌

3

4

,‌

1

4

)