UPSC NDA I 2025 Mathematics Paper

Given,
The circle's equation is
x2+y2+2x+2y+1=0
Rewrite by completing squares:
(x+1)2+(y+1)2=1
∴ center =(−1,−1), radius r=1
For a square inscribed in this circle with sides parallel to the axes, its diagonal equals the circle's diameter =2. If the square has side length a, then
a√‌2=2⇒a=√‌2.
Each vertex lies a half-side =‌

a

2

=‌

1

√2

from the center along both axes. Since the center is (−1,−1), the four vertices are
(−1±‌

1

√2

,−1±‌

1

√2

).
One of these vertices is
(−1+‌

1

√2

,−1−‌

1

√2

).