TS EAMCET 6-Aug-2021 Shift 2 Question Paper

Given, system of family of circles is
x2+y2+2fy+λ(x2+y2+2gx+k)‌‌=0
⇒‌‌(1+λ)x2+(1+λ)y2+2gλx+2fy+λk‌‌=0
⇒‌‌x2+y2+2⋅(‌

gλ

1+λ

)x+2(‌

f

1+λ

)y+‌

λk

1+λ

=0
For point circle, radius should be zero.
‌∴‌‌‌

g2λ2

(1+λ)2

+‌

f2

(1+λ)2

=‌

λk

1+λ

⇒‌g2λ2+f2=k(λ+λ2)
⇒(g2−k)λ2−kλ+f2=0 . . . (i)
Let the quadratic Eq. (i) has roots λ1 and λ2, then the coordinates of point circles are c1(‌

−gλ1

1+λ1

,‌

f

1+λ1

) and c2(‌

−gλ2

1+λ2

,‌

f

1+λ2

)
∵ Lines OC1 and OC2 are perpendicular to each other. So, ‌‌‌

‌ f 1+λ2

gλ2 1+λ2

×‌

‌ f 1+λ1

gλ1 1+λ1

=−1 . . . (ii)
∵ Product of roots of Eq. (i), we get
λ1λ2=‌

f2

g2−k

. . . (iii)
From Eqs. (ii) and (iii), we get
⇒‌‌g2=−g2+k
⇒2g2=k⇒g2=‌

k

2