UPSC NDA I 2025 Mathematics Paper

Given,
Hyperbola equation: 25x2−75y2=225
Divide both sides by 225 to obtain standard form:
‌

25x2

225

−‌

75y2

225

=1⟹‌

x2

9

−‌

y2

3

=1
Thus, a2=9 and b2=3.
Compute c from c2=a2+b2 :
c2=9+3=12⇒c=√‌12=2√‌3
The foci are at (±c,0), so the distance between them is 2 c :
2c=2×2√‌3=4√‌3
∴ The distance between the two foci is 4√‌3 units.