`x^2 +y^2 = 16 and x^2 +y^2=36` are two circles. If `P and Q` move respectively on these circles such that `PQ=4` then the locus of mid-point of `PQ` is a circle of radius

Options

A. `sqrt(20)`
B. `sqrt(22)`
C. `sqrt(30)`
D. `sqrt(32)`

Explanation

To find the locus of the midpoint of segment \( PQ \) where \( P \) and \( Q \) lie on the circles defined by the equations \( x^2 + y^2 = 16 \) and \( x^2 + y^2 = 36 \) respectively, and where the distance \( PQ = 4 \), we can follow these steps:

### Step 1: Identify the centers and radii of the circles
The first circle \( x^2 + y^2 = 16 \) has:
- Center: \( (0, 0) \)
- Radius: \( r_1 = \sqrt{16} = 4 \)

The second circle \( x^2 + y^2 = 36 \) has:
- Center: \( (0, 0) \)
- Radius: \( r_2 = \sqrt{36} = 6 \)

### Step 2: Define points \( P \) and \( Q \)
Let:
- \( P(x_1, y_1) \) be a point on the first circle, so \( x_1^2 + y_1^2 = 16 \)
- \( Q(x_2, y_2) \) be a point on the second circle, so \( x_2^2 + y_2^2 = 36 \)

### Step 3: Use the distance condition
The distance \( PQ = 4 \) implies:
\[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = 4 \]
Squaring both sides gives:
\[ (x_2 - x_1)^2 + (y_2 - y_1)^2 = 16 \]

### Step 4: Find the midpoint \( R \) of \( PQ \)
The coordinates of the midpoint \( R \) are:
\[ R\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

### Step 5: Express \( x_1, y_1, x_2, y_2 \) in terms of \( R \)
Let:
- \( x = \frac{x_1 + x_2}{2} \)
- \( y = \frac{y_1 + y_2}{2} \)

Then:
\[ x_1 = x - a, \quad y_1 = y - b \]
\[ x_2 = x + a, \quad y_2 = y + b \]
where \( a \) and \( b \) are the half differences of \( x_1, x_2 \) and \( y_1, y_2 \) respectively.

### Step 6: Substitute into the circle equations
Substituting \( x_1 \) and \( y_1 \) into the first circle's equation:
\[ (x - a)^2 + (y - b)^2 = 16 \]
Expanding gives:
\[ x^2 - 2ax + a^2 + y^2 - 2by + b^2 = 16 \]

Substituting \( x_2 \) and \( y_2 \) into the second circle's equation:
\[ (x + a)^2 + (y + b)^2 = 36 \]
Expanding gives:
\[ x^2 + 2ax + a^2 + y^2 + 2by + b^2 = 36 \]

### Step 7: Add the two equations
Adding both equations:
\[ (x^2 - 2ax + a^2 + y^2 - 2by + b^2) + (x^2 + 2ax + a^2 + y^2 + 2by + b^2) = 16 + 36 \]
This simplifies to:
\[ 2x^2 + 2y^2 + 2a^2 + 2b^2 = 52 \]
Dividing by 2 gives:
\[ x^2 + y^2 + a^2 + b^2 = 26 \]

### Step 8: Relate \( a^2 + b^2 \) to the distance \( PQ \)
Since \( PQ = 4 \), we have:
\[ a^2 + b^2 = 16 \]
Thus:
\[ x^2 + y^2 + 16 = 26 \]
This leads to:
\[ x^2 + y^2 = 10 \]

### Conclusion
The locus of the midpoint \( R \) is a circle with the equation:
\[ x^2 + y^2 = 10 \]
This means the radius \( r \) of the locus circle is:
\[ r = \sqrt{10} \]

### Final Answer
The radius of the locus of the midpoint \( R \) is \( \sqrt{10} \).

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