`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 `X^(o+)` and `Y^(Θ)`, respectively, are

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 find the ionic radii of \(X^{2-}\) and \(Y^{2-}\) for the compounds \(AX\) and \(AY\) with rock salt type structure, we can follow these steps:

### Step 1: Determine the Edge Lengths
For the rock salt structure, the anion-anion distance is related to the edge length \(a\) of the cubic unit cell. The relationship is given by:
\[ \text{Anion-Anion Distance} = \frac{a}{\sqrt{2}} \]
For \(AX\), the anion-anion distance is 2.40 Å:
\[ 2.40 = \frac{a_1}{\sqrt{2}} \implies a_1 = 2.40 \times \sqrt{2} = 2.40 \times 1.414 \approx 3.4 \, \text{Å} \]

For \(AY\), the anion-anion distance is 1.63 Å:
\[ 1.63 = \frac{a_2}{\sqrt{2}} \implies a_2 = 1.63 \times \sqrt{2} = 1.63 \times 1.414 \approx 2.30 \, \text{Å} \]

### Step 2: Calculate Cation-Anion Distances
The cation-anion distance is given by:
\[ \text{Cation-Anion Distance} = \frac{a}{2} \]
For \(AX\), the cation-anion distance is 1.70 Å:
\[ 1.70 = \frac{a_1}{2} \implies a_1 = 1.70 \times 2 = 3.4 \, \text{Å} \quad \text{(consistent with previous calculation)} \]

For \(AY\), the cation-anion distance is 1.15 Å:
\[ 1.15 = \frac{a_2}{2} \implies a_2 = 1.15 \times 2 = 2.30 \, \text{Å} \quad \text{(consistent with previous calculation)} \]

### Step 3: Establish the Relationship Between Ionic Radii
In a rock salt structure, the ratio of the cation radius \(R^+\) to the anion radius \(R^-\) is given by:
\[ \frac{R^+}{R^-} = 0.414 \]
Thus, we can express the cation radius in terms of the anion radius:
\[ R^+ = 0.414 R^- \]

### Step 4: Set Up the Equation for AX
For \(AX\):
\[ R^+ + R^- = 1.70 \, \text{Å} \]
Substituting \(R^+\):
\[ 0.414 R^- + R^- = 1.70 \]
\[ (0.414 + 1) R^- = 1.70 \]
\[ 1.414 R^- = 1.70 \implies R^- = \frac{1.70}{1.414} \approx 1.20 \, \text{Å} \]
Thus, the ionic radius of \(X^{2-}\) is approximately \(1.20 \, \text{Å}\).

### Step 5: Set Up the Equation for AY
For \(AY\):
\[ R^+ + R^- = 1.15 \, \text{Å} \]
Substituting \(R^+\):
\[ 0.414 R^- + R^- = 1.15 \]
\[ (0.414 + 1) R^- = 1.15 \]
\[ 1.414 R^- = 1.15 \implies R^- = \frac{1.15}{1.414} \approx 0.81 \, \text{Å} \]
Thus, the ionic radius of \(Y^{2-}\) is approximately \(0.8 \, \text{Å}\).

### Final Results
- Ionic radius of \(X^{2-}\) is approximately \(1.20 \, \text{Å}\).
- Ionic radius of \(Y^{2-}\) is approximately \(0.8 \, \text{Å}\).