`AX,AY,BX`, and `BY` have rock salt type structure with following internuclear distances:
`|{:("Salt","Anion-anion","Cation-anion"),(,"distance in"Å,"distance in"Å),(AX,2.40,1.70),(AY,1.63,1.15),(BX,2.66,1.88),(By,2.09,1.48):}|`
Ionic radii of `A^(o+)` and `B^(o+)`, respectively, are

a.`0.35` and `0.68 Å`
b.`0.68` and `0.35 Å`
c.`1.20` and `0.80 Å`
d.`0.80` and `1.20 Å`

Options

A. `0.35` and `0.68 Å`
B. `0.68` and `0.35 Å`
C. `1.20` and `0.80 Å`
D. `0.80` and `1.20 Å`

Explanation

To determine the ionic radii of \( A^{+} \) and \( B^{+} \) cations in the rock salt type structure, we will use the given inter-nuclear distances of the anions and cations. The distances provided are:

- For AX:
- Anion-anion distance (X-X) = 2.40 Å
- Cation-anion distance (A-X) = 1.70 Å

- For AY:
- Anion-anion distance (Y-Y) = 1.63 Å
- Cation-anion distance (A-Y) = 1.15 Å

- For BX:
- Anion-anion distance (X-X) = 2.66 Å
- Cation-anion distance (B-X) = 1.88 Å

- For BY:
- Anion-anion distance (Y-Y) = 2.09 Å
- Cation-anion distance (B-Y) = 1.48 Å

### Step 1: Calculate the radius of the anion \( R^{-} \)

Using the anion-anion distance for \( AX \):
\[ R_{X^{-}} = \frac{\text{Anion-Anion Distance (X-X)}}{2} = \frac{2.40 \, \text{Å}}{2} = 1.20 \, \text{Å} \]

### Step 2: Calculate the radius of cation \( B^{+} \)

Using the cation-anion distance for \( BX \):
\[ R_{B^{+}} = \text{Cation-Anion Distance (B-X)} - R_{X^{-}} = 1.88 \, \text{Å} - 1.20 \, \text{Å} = 0.68 \, \text{Å} \]

### Step 3: Calculate the radius of cation \( A^{+} \)

Using the cation-anion distance for \( AY \):
\[ R_{A^{+}} = \text{Cation-Anion Distance (A-Y)} - R_{Y^{-}} \]
First, we need to find \( R_{Y^{-}} \) using the anion-anion distance for \( AY \):
\[ R_{Y^{-}} = \frac{\text{Anion-Anion Distance (Y-Y)}}{2} = \frac{1.63 \, \text{Å}}{2} = 0.815 \, \text{Å} \]
Now substituting this back:
\[ R_{A^{+}} = 1.15 \, \text{Å} - 0.815 \, \text{Å} = 0.335 \, \text{Å} \approx 0.35 \, \text{Å} \]

### Final Result

The ionic radii of \( A^{+} \) and \( B^{+} \) are:
- \( R_{A^{+}} = 0.35 \, \text{Å} \)
- \( R_{B^{+}} = 0.68 \, \text{Å} \)

Thus, the correct option is **a. \( 0.35 \) and \( 0.68 \, \text{Å} \)**.

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