NDA 2 2025 Maths, question 53

Topic: Analytical geometry

Passage · Q53 to Q54

Suppose $S$ is the sphere with the smallest radius that passes through the points $A(1,0,0)$ , $B(0,1,0)$ and $C(0,0,1)$ .

Q53

What is the radius of $S$ ?

(a) $\sqrt{\frac{1}{3}}$
(b) $\sqrt{\frac{2}{3}}$ Correct answer
(c) $\frac{1}{3}$
(d) $1$

Solution.

Answer: (b) $\sqrt{\frac{2}{3}}$

Equidistance from A,B,C requires the sphere centre to have coordinates (t,t,t).
Its radius squared is (t−1)²+t²+t²=3(t−1/3)²+2/3, minimized at t=1/3.
Thus the smallest radius is √(2/3).