Let ABCD be a square of side length 2 units. Let `C_1` be the incircle and `C_2` be the circumcircle of the square ABCD. If P is a point on `C_1` and Q is a point on `C_2`, the value of `(PA^2+PB^2+PC^2+PD^2)/(QA^2+QB^2+QC^2+QD^2)` is

Options

A. A) 0.75
B. B) 1.25
C. C) 1
D. D) 0.5

Answer

To solve the problem, we need to calculate the values of \( PA^2 + PB^2 + PC^2 + PD^2 \) for point \( P \) on the incircle \( C_1 \) and \( QA^2 + QB^2 + QC^2 + QD^2 \) for point \( Q \) on the circumcircle \( C_2 \) of square \( ABCD \).

### Step 1: Define the square and its properties
Let the square \( ABCD \) have vertices:
- \( A(0, 0) \)
- \( B(2, 0) \)
- \( C(2, 2) \)
- \( D(0, 2) \)

The side length of the square is 2 units.

### Step 2: Calculate the radius of the incircle \( C_1 \)
The radius of the incircle \( C_1 \) is half the side length of the square:
\[ r = \frac{2}{2} = 1 \text{ unit} \]

### Step 3: Calculate the radius of the circumcircle \( C_2 \)
The radius of the circumcircle \( C_2 \) is half the length of the diagonal of the square. The diagonal \( d \) can be calculated using the Pythagorean theorem:
\[ d = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2} \]
Thus, the radius \( R \) of the circumcircle is:
\[ R = \frac{d}{2} = \sqrt{2} \text{ units} \]

### Step 4: Choose points \( P \) and \( Q \)
Let \( P \) be the midpoint of side \( CD \), which is on the incircle:
\[ P(1, 2) \]
Let \( Q \) be point \( D \), which is a vertex of the square and lies on the circumcircle:
\[ Q(0, 2) \]

### Step 5: Calculate \( PA^2 + PB^2 + PC^2 + PD^2 \)
Now we calculate the distances from point \( P(1, 2) \) to each vertex:
1. \( PA^2 = (1 - 0)^2 + (2 - 0)^2 = 1 + 4 = 5 \)
2. \( PB^2 = (1 - 2)^2 + (2 - 0)^2 = 1 + 4 = 5 \)
3. \( PC^2 = (1 - 2)^2 + (2 - 2)^2 = 1 + 0 = 1 \)
4. \( PD^2 = (1 - 0)^2 + (2 - 2)^2 = 1 + 0 = 1 \)

Now, summing these values:
\[ PA^2 + PB^2 + PC^2 + PD^2 = 5 + 5 + 1 + 1 = 12 \]

### Step 6: Calculate \( QA^2 + QB^2 + QC^2 + QD^2 \)
Now we calculate the distances from point \( Q(0, 2) \) to each vertex:
1. \( QA^2 = (0 - 0)^2 + (2 - 0)^2 = 0 + 4 = 4 \)
2. \( QB^2 = (0 - 2)^2 + (2 - 0)^2 = 4 + 4 = 8 \)
3. \( QC^2 = (0 - 2)^2 + (2 - 2)^2 = 4 + 0 = 4 \)
4. \( QD^2 = (0 - 0)^2 + (2 - 2)^2 = 0 + 0 = 0 \)

Now, summing these values:
\[ QA^2 + QB^2 + QC^2 + QD^2 = 4 + 8 + 4 + 0 = 16 \]

### Step 7: Calculate the final ratio
Now we can compute the ratio:
\[ \frac{PA^2 + PB^2 + PC^2 + PD^2}{QA^2 + QB^2 + QC^2 + QD^2} = \frac{12}{16} = \frac{3}{4} \]

### Final Answer
The value of \( \frac{PA^2 + PB^2 + PC^2 + PD^2}{QA^2 + QB^2 + QC^2 + QD^2} \) is \( \frac{3}{4} \).