D

Step 1: Determine the mass of the muon and the nucleus.
The mass of the muon is given as \( m_\mu = 207 m_e \). The mass of the proton is given as \( m_p = 1836 m_e \).

The effective mass for the muonic atom can be approximated to be the mass of the proton since it's significantly larger than the muon, i.e., \( M = m_p \approx 1836 m_e \).

Step 2: Calculate the radius of the first Bohr orbit.
The radius of the first Bohr orbit for a two-body system can be given by the formula:
\[ r = \frac{n^2 h^2}{4 \pi^2 k e^2} \cdot \frac{1}{M} \]
For the first orbit (principal quantum number \( n = 1 \)):
\( h \) is Planck's constant, \( k \) is Coulomb's constant, and \( e \) is the electron charge.
Therefore, the radius will be roughly:
\[ r = \frac{(1^2) h^2}{4 \pi^2 k e^2} \cdot \frac{1}{m_p} \]
Using the approximations that incorporate the effective mass, we find the radius is approximately:
\[ r = \frac{h^2}{4 \pi^2 k e^2 M} \approx \frac{0.529 \times 1836}{207} \approx 0.00024 \text{ m} \text{ or } 2.4 \times 10^{-4} \text{ m} \text{ (very rough estimation)} \]

Step 3: Calculate the ionization energy.
The ionization energy for a hydrogen-like atom is given by:
\[ E = \frac{Z^2 \mu e^4}{8 \varepsilon_0^2 h^2} \] where \( Z \) is the atomic number (1 for hydrogen), and \( \mu \) is the reduced mass:
\[ \mu = \frac{m_p m_\mu}{m_p + m_\mu} \approx \frac{1836 imes 207}{1836 + 207} m_e \approx 195.6 m_e \] (a simplified estimate).
Substituting the values yields:
\[ E = \frac{(1^2)(195.6 m_e e^4)}{8 (8.854 \times 10^{-12})^2 h^2} \approx 1.88 imes 10^{-13} \text{ J} \approx 1.17 \text{ MeV} \]

Thus, the final results are roughly:
Radius of the first Bohr orbit: \( 2.4 \times 10^{-4} \, ext{m} \) and Ionization energy: \( 1.17 ext{ MeV} \).
Picked option from the given choices corresponds to March calculation of ionization energy.