Let the line segment joining the centres of the circles `x^(2)-2x+y^(2)=0` and `x^(2)+y^(2)+4x+8y+16=0` intersect the circles at P and Q respectively. Then the equation of the circle with PQ as its diameter is

Options

A. `5^(2)+5y^(2)-2x-16y+8=0`
B. `5x^(2)+5y^(2)-8x-24y+27=0`
C. `5x^(2)+5y^(2)+8x+24y+27=0`
D. `5x^(2)+5y^(2)+2x+16y+8=0`

Explanation

To solve the problem, we need to find the equation of the circle with diameter PQ, where P and Q are the points where the line segment joining the centers of two given circles intersects the circles.

### Step 1: Identify the centers and radii of the given circles.

1. **Circle 1**: The equation is \(x^2 - 2x + y^2 = 0\).
- Rewriting it: \((x - 1)^2 + y^2 = 1\).
- Center \(C_1 = (1, 0)\) and radius \(r_1 = 1\).

2. **Circle 2**: The equation is \(x^2 + y^2 + 4x + 8y + 16 = 0\).
- Rewriting it: \((x + 2)^2 + (y + 4)^2 = 4\).
- Center \(C_2 = (-2, -4)\) and radius \(r_2 = 2\).

### Step 2: Find the distance between the centers \(C_1\) and \(C_2\).

- The distance \(d\) between \(C_1\) and \(C_2\) is calculated as:
\[ d = \sqrt{((-2) - 1)^2 + ((-4) - 0)^2} = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5. \]

### Step 3: Find the coordinates of points P and Q.

Since the distance between the centers is 5, and the radius of circle 1 is 1 and that of circle 2 is 2, we can find the coordinates of points P and Q.

- The line segment \(C_1C_2\) can be divided in the ratio of their respective radii, which is \(1:2\).

Using the section formula, the coordinates of point \(C_3\) (the midpoint of PQ) can be calculated as:
\[ C_3 = \left( \frac{2 \cdot (-2) + 3 \cdot 1}{2 + 3}, \frac{2 \cdot (-4) + 3 \cdot 0}{2 + 3} \right) = \left( \frac{-4 + 3}{5}, \frac{-8 + 0}{5} \right) = \left( -\frac{1}{5}, -\frac{8}{5} \right). \]

### Step 4: Find the equation of the circle with diameter PQ.

The center of the circle with diameter PQ is at \(C_3\) and the radius is half the distance \(PQ\).

- The distance \(PQ\) is equal to the distance between the centers \(C_1\) and \(C_2\) minus the sum of the radii:
\[ PQ = d - (r_1 + r_2) = 5 - (1 + 2) = 2. \]
Thus, the radius of the circle with diameter PQ is \(1\).

### Step 5: Write the equation of the circle.

The standard form of the equation of a circle is given by:
\[ (x - h)^2 + (y - k)^2 = r^2, \]
where \((h, k)\) is the center and \(r\) is the radius.

Substituting \(h = -\frac{1}{5}\), \(k = -\frac{8}{5}\), and \(r = 1\):
\[ \left(x + \frac{1}{5}\right)^2 + \left(y + \frac{8}{5}\right)^2 = 1. \]

### Step 6: Simplifying the equation.

Expanding this:
\[ \left(x + \frac{1}{5}\right)^2 + \left(y + \frac{8}{5}\right)^2 = 1 \]
gives:
\[ x^2 + \frac{2}{5}x + \frac{1}{25} + y^2 + \frac{16}{5}y + \frac{64}{25} = 1. \]
Combining terms:
\[ x^2 + y^2 + \frac{2}{5}x + \frac{16}{5}y + \frac{65}{25} - 1 = 0. \]
This simplifies to:
\[ x^2 + y^2 + \frac{2}{5}x + \frac{16}{5}y + \frac{40}{25} = 0. \]

### Final Equation

Multiplying through by 25 to eliminate fractions:
\[ 25x^2 + 25y^2 + 10x + 80y + 40 = 0. \]

### Conclusion

Thus, the equation of the circle with diameter PQ is:
\[ 5x^2 + 5y^2 + 2x + 16y + 8 = 0. \]