AP EAPCET 24 May 2025 Shift 1 Paper

s1:x2+y2+2gx+2fy+c=0
s2:2x2+2y2+3x+8y+2c=0
⇒‌‌x2+y2+‌

3

2

x+4y+c=0
s3:x2+y2+2x+2y+1=0
Radical axis s1−s2=0
⇒(x2+y2+2gx+2fy+c)
‌−(x2+y2+‌

3

2

x+4y+c)+0
⇒2gx+2fy−‌

3x

2

−4y=0
⇒(2g−‌

3

2

)x+(2f−4)y=0
⇒(4g−3)x+(4f−8)y=0
The radical axis touches the circle s3, where radius of s3.
=√(−1)2+(−1)2−1=1‌ and center is ‌(−1,−1)
The perpendicular distance from the center of s3 to the radical axis = radius of s3
‌‌

|(4g−3)⋅(−1)+(4f−8)(−1)|

√(4g−3)2+(4f−8)2

=1
‌⇒‌‌‌

|−4g+3−4f+8|

√(4g−3)2+(4f−8)2

=1
‌⇒‌‌|−4g−4f+11|
‌=√(4g−3)2+(4f−8)2
‌⇒‌‌(−4g−4f+11)2
‌=(4g−3)2+(4f−8)2
‌⇒‌‌16g2+16f2+121+32gf−88g−88f
‌=16g2−24g+9+16f2−64f+64
‌⇒‌‌32gf−64g−24f+48=0
‌⇒‌‌8gf−16g−6f+12=0
‌⇒‌‌8g(f−2)−6(f−2)=0
‌⇒‌‌(8g−6)(f−2)=0
‌⇒‌‌4g−3)(f−2)=0
‌⇒‌‌g=‌

3

4

‌ or ‌f=2
Either g=‌

3

4

or f=2 is the required condition.