(i) Find the equation of a circle , which is concentric with the circle `x^(2) + y^(2) - 6x + 12y + 15 = 0` and of double its radius.
(ii) Find the equation of a circle , which is concentric with the circle `x^(2) + y^(2) - 2x - 4y + 1 = 0` and whose radius is 5.
(iii) Find the equation of the cricle concentric with `x^(2) + y^(2) - 4x - 6y - 3 = 0` and which touches the y-axis.
(iv) find the equation of a circle passing through the centre of the circle `x^(2) + y^(2) + 8x + 10y - 7 = 0 ` and concentric with the circle
`2x^(2) + 2y^(2) - 8x - 12y - 9 = 0`.
(v) Find the equation of the circle concentric with the circle `x^(2) + y^(2) + 4x - 8y - 6 = 0 ` and having radius double of its radius.

Answer

To solve the given problem step by step, we will tackle each part of the question systematically.

### Part (i)
**Find the equation of a circle which is concentric with the circle \(x^2 + y^2 - 6x + 12y + 15 = 0\) and of double its radius.**

1. **Identify the given circle's equation**:
\[ x^2 + y^2 - 6x + 12y + 15 = 0 \]

2. **Rewrite in standard form**:
\[ (x^2 - 6x) + (y^2 + 12y) + 15 = 0 \]
Completing the square for \(x\) and \(y\):
\[ (x - 3)^2 - 9 + (y + 6)^2 - 36 + 15 = 0 \]
This simplifies to:
\[ (x - 3)^2 + (y + 6)^2 = 30 \]
The center is \((3, -6)\) and the radius \(R_1 = \sqrt{30}\).

3. **Find the new radius**:
The new radius \(R_2\) is double the original radius:
\[ R_2 = 2R_1 = 2\sqrt{30} \]

4. **Write the equation of the new circle**:
The equation is:
\[ (x - 3)^2 + (y + 6)^2 = (2\sqrt{30})^2 = 120 \]
Expanding this gives:
\[ x^2 - 6x + 9 + y^2 + 12y + 36 = 120 \]
Simplifying:
\[ x^2 + y^2 - 6x + 12y - 75 = 0 \]

### Part (ii)
**Find the equation of a circle which is concentric with the circle \(x^2 + y^2 - 2x - 4y + 1 = 0\) and whose radius is 5.**

1. **Identify the given circle's equation**:
\[ x^2 + y^2 - 2x - 4y + 1 = 0 \]

2. **Rewrite in standard form**:
Completing the square:
\[ (x^2 - 2x) + (y^2 - 4y) + 1 = 0 \]
This gives:
\[ (x - 1)^2 - 1 + (y - 2)^2 - 4 + 1 = 0 \]
Simplifying:
\[ (x - 1)^2 + (y - 2)^2 = 4 \]
The center is \((1, 2)\) and the radius is \(R = 2\).

3. **Write the equation of the new circle**:
The new radius is 5:
\[ (x - 1)^2 + (y - 2)^2 = 5^2 = 25 \]
Expanding gives:
\[ x^2 - 2x + 1 + y^2 - 4y + 4 = 25 \]
Simplifying:
\[ x^2 + y^2 - 2x - 4y - 20 = 0 \]

### Part (iii)
**Find the equation of the circle concentric with \(x^2 + y^2 - 4x - 6y - 3 = 0\) and which touches the y-axis.**

1. **Identify the given circle's equation**:
\[ x^2 + y^2 - 4x - 6y - 3 = 0 \]

2. **Rewrite in standard form**:
Completing the square:
\[ (x^2 - 4x) + (y^2 - 6y) - 3 = 0 \]
This gives:
\[ (x - 2)^2 - 4 + (y - 3)^2 - 9 - 3 = 0 \]
Simplifying:
\[ (x - 2)^2 + (y - 3)^2 = 16 \]
The center is \((2, 3)\) and the radius is \(R = 4\).

3. **Determine the radius to touch the y-axis**:
Since it touches the y-axis, the radius is equal to the x-coordinate of the center:
\[ R = 2 \]

4. **Write the equation of the new circle**:
\[ (x - 2)^2 + (y - 3)^2 = 2^2 = 4 \]
Expanding gives:
\[ x^2 - 4x + 4 + y^2 - 6y + 9 = 4 \]
Simplifying:
\[ x^2 + y^2 - 4x - 6y + 9 = 0 \]

### Part (iv)
**Find the equation of a circle passing through the center of the circle \(x^2 + y^2 + 8x + 10y - 7 = 0\) and concentric with the circle \(2x^2 + 2y^2 - 8x - 12y - 9 = 0\).**

1. **Identify the first circle's equation**:
\[ x^2 + y^2 + 8x + 10y - 7 = 0 \]

2. **Rewrite in standard form**:
Completing the square:
\[ (x^2 + 8x) + (y^2 + 10y) - 7 = 0 \]
This gives:
\[ (x + 4)^2 - 16 + (y + 5)^2 - 25 - 7 = 0 \]
Simplifying:
\[ (x + 4)^2 + (y + 5)^2 = 48 \]
The center is \((-4, -5)\).

3. **Identify the second circle's equation**:
\[ 2x^2 + 2y^2 - 8x - 12y - 9 = 0 \implies x^2 + y^2 - 4x - 6y - \frac{9}{2} = 0 \]

4. **Rewrite in standard form**:
Completing the square:
\[ (x^2 - 4x) + (y^2 - 6y) - \frac{9}{2} = 0 \]
This gives:
\[ (x - 2)^2 - 4 + (y - 3)^2 - 9 - \frac{9}{2} = 0 \]
Simplifying:
\[ (x - 2)^2 + (y - 3)^2 = \frac{25}{2} \]
The center is \((2, 3)\).

5. **Write the equation of the new circle**:
The new circle must pass through \((-4, -5)\) and be concentric with the center \((2, 3)\):
\[ (x - 2)^2 + (y - 3)^2 = R^2 \]
To find \(R\), we calculate:
\[ R^2 = (-4 - 2)^2 + (-5 - 3)^2 = 36 + 64 = 100 \]
Thus:
\[ (x - 2)^2 + (y - 3)^2 = 100 \]
Expanding gives:
\[ x^2 - 4x + 4 + y^2 - 6y + 9 = 100 \]
Simplifying:
\[ x^2 + y^2 - 4x - 6y - 87 = 0 \]

### Part (v)
**Find the equation of a circle concentric with the circle \(x^2 + y^2 + 4x - 8y - 6 = 0\) and having radius double of its radius.**

1. **Identify the given circle's equation**:
\[ x^2 + y^2 + 4x - 8y - 6 = 0 \]

2. **Rewrite in standard form**:
Completing the square:
\[ (x^2 + 4x) + (y^2 - 8y) - 6 = 0 \]
This gives:
\[ (x + 2)^2 - 4 + (y - 4)^2 - 16 - 6 = 0 \]
Simplifying:
\[ (x + 2)^2 + (y - 4)^2 = 26 \]
The center is \((-2, 4)\) and the radius is \(R = \sqrt{26}\).

3. **Find the new radius**:
The new radius \(R_2\) is double the original radius:
\[ R_2 = 2R = 2\sqrt{26} \]

4. **Write the equation of the new circle**:
\[ (x + 2)^2 + (y - 4)^2 = (2\sqrt{26})^2 = 104 \]
Expanding gives:
\[ x^2 + 4x + 4 + y^2 - 8y + 16 = 104 \]
Simplifying:
\[ x^2 + y^2 + 4x - 8y - 84 = 0 \]

### Summary of Solutions
1. \(x^2 + y^2 - 6x + 12y - 75 = 0\)
2. \(x^2 + y^2 - 2x - 4y - 20 = 0\)
3. \(x^2 + y^2 - 4x - 6y + 9 = 0\)
4. \(x^2 + y^2 - 4x - 6y - 87 = 0\)
5. \(x^2 + y^2 + 4x - 8y - 84 = 0\)