1. Bohr radius for hydrogen-like ions

   For a hydrogen-like ion, the radius of the $n^{\text{th}}$ orbit is $r_n = \frac{n^2 a_0}{Z}$ where:

   • $a_0$ = first Bohr radius for hydrogen,
   • $Z$ = atomic number,
   • $n$ = principal quantum number.

2. For $\mathrm{Li}^{++}$ ion

   Lithium has atomic number $Z = 3$

   Since it is in the ground state, $n = 1$

3. Substitute into the formula

   $r_1 = \frac{1^2 a_0}{3} = \frac{a_0}{3}$

4. Compare with the given form

   Given: $r = \frac{1}{X} a_0$

   So, $\frac{1}{X} a_0 = \frac{a_0}{3}$

   Therefore, $X = 3$

5. Option check

   • A: $2$ ❌
   • B: $9$ ❌
   • C: $1$ ❌
   • D: $3$ ✅

Final Answer: $X = 3$