TG EAMCET 2 May 2025 Shift 2 Paper

Given, equation of circles
x2+y2=16
And (x−9)2+y2=16
C1‌=(0,0),r1=4
C2‌=(9,0),r2=4
∴‌‌C1C2‌=√(9−0)2+(0−0)2
‌=√(9)2=9
And r1+r2=4+4=8
Since C1C2>r1+r2
Circles are separate to each other, any tangent to circle (i) with slope ( m ) is
‌y=mx±r√1+m2
⇒y=mx+4√1+m2
⇒mx−y±4√1+m2=0
Equation (iii) to be tangent to circle (ii)
∴‌|‌

m(9)−0±4√1+m2

√m2+1

|=4
⇒16(m2+1)=(9m±4√1+m2)2
⇒16m2+16=81m2+16(1+m2)
‌±72m√1+m2
⇒81m2±72m√1+m2=0
⇒9m(9m±8√1+m2)=0
When 9m=0⇒m=0
Then Eq. (iii) y=±4 (horizontal lines)
If, 9m±8√1+m2=0
‌⇒‌‌(9m)2=(±8√1+m2)2
‌⇒‌‌81m2=64(1+m2)
‌⇒‌‌81m2−64m2=64
‌⇒‌‌17m2=64
‌⇒‌‌m=±√‌

64

17

‌⇒‌‌m=±‌

8

√17