Find the centre and radius of each of the circles whose equations are given below:
(i) `3x^(2)+3y^(2)-5x-6y+4=0`
(ii) `3x^(2)+3y^(2)+6x-12-1=0`
(iii) `x^(2)+y^(2)+6x+8y-96=0`
(iv) `2x^(2)+2y^(2)-4x+6y-3=0`
(v) `2x^(2)+2y^(2)-3x+2y-1=0`
(vi)` x^(2)+y^(2)+2x-4y-4=0`
(vii) `x^(2)+y^(2)-4x-8y-41=0`

Answer

To find the center and radius of the circles given by the equations, we will follow these steps for each equation:

1. **Rewrite the equation in standard form**: The general equation of a circle is given by:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
where the center is \((-g, -f)\) and the radius is \(\sqrt{g^2 + f^2 - c}\).

2. **Identify coefficients**: We will identify the coefficients \(g\), \(f\), and \(c\) from the rewritten equation.

3. **Calculate the center and radius**: Using the identified coefficients, we will calculate the center and radius.

Now, let's solve each equation step by step:

### (i) \(3x^2 + 3y^2 - 5x - 6y + 4 = 0\)

1. **Divide the entire equation by 3**:
\[ x^2 + y^2 - \frac{5}{3}x - 2y + \frac{4}{3} = 0 \]

2. **Identify coefficients**:
- \(2g = -\frac{5}{3} \Rightarrow g = -\frac{5}{6}\)
- \(2f = -2 \Rightarrow f = -1\)
- \(c = \frac{4}{3}\)

3. **Calculate center and radius**:
- Center: \((-g, -f) = \left(\frac{5}{6}, 1\right)\)
- Radius: \(\sqrt{g^2 + f^2 - c} = \sqrt{\left(-\frac{5}{6}\right)^2 + (-1)^2 - \frac{4}{3}} = \sqrt{\frac{25}{36} + 1 - \frac{4}{3}} = \sqrt{\frac{25}{36} + \frac{36}{36} - \frac{48}{36}} = \sqrt{\frac{13}{36}} = \frac{\sqrt{13}}{6}\)

### (ii) \(3x^2 + 3y^2 + 6x - 12y - 1 = 0\)

1. **Divide the entire equation by 3**:
\[ x^2 + y^2 + 2x - 4y - \frac{1}{3} = 0 \]

2. **Identify coefficients**:
- \(2g = 2 \Rightarrow g = 1\)
- \(2f = -4 \Rightarrow f = -2\)
- \(c = -\frac{1}{3}\)

3. **Calculate center and radius**:
- Center: \((-g, -f) = (-1, 2)\)
- Radius: \(\sqrt{g^2 + f^2 - c} = \sqrt{1^2 + (-2)^2 - (-\frac{1}{3})} = \sqrt{1 + 4 + \frac{1}{3}} = \sqrt{5 + \frac{1}{3}} = \sqrt{\frac{15}{3} + \frac{1}{3}} = \sqrt{\frac{16}{3}} = \frac{4}{\sqrt{3}}\)

### (iii) \(x^2 + y^2 + 6x + 8y - 96 = 0\)

1. **Identify coefficients**:
- \(2g = 6 \Rightarrow g = 3\)
- \(2f = 8 \Rightarrow f = 4\)
- \(c = -96\)

2. **Calculate center and radius**:
- Center: \((-g, -f) = (-3, -4)\)
- Radius: \(\sqrt{g^2 + f^2 - c} = \sqrt{3^2 + 4^2 - (-96)} = \sqrt{9 + 16 + 96} = \sqrt{121} = 11\)

### (iv) \(2x^2 + 2y^2 - 4x + 6y - 3 = 0\)

1. **Divide the entire equation by 2**:
\[ x^2 + y^2 - 2x + 3y - \frac{3}{2} = 0 \]

2. **Identify coefficients**:
- \(2g = -2 \Rightarrow g = -1\)
- \(2f = 3 \Rightarrow f = \frac{3}{2}\)
- \(c = -\frac{3}{2}\)

3. **Calculate center and radius**:
- Center: \((-g, -f) = (1, -\frac{3}{2})\)
- Radius: \(\sqrt{g^2 + f^2 - c} = \sqrt{(-1)^2 + \left(\frac{3}{2}\right)^2 - (-\frac{3}{2})} = \sqrt{1 + \frac{9}{4} + \frac{3}{2}} = \sqrt{1 + \frac{9}{4} + \frac{6}{4}} = \sqrt{\frac{19}{4}} = \frac{\sqrt{19}}{2}\)

### (v) \(2x^2 + 2y^2 - 3x + 2y - 1 = 0\)

1. **Divide the entire equation by 2**:
\[ x^2 + y^2 - \frac{3}{2}x + y - \frac{1}{2} = 0 \]

2. **Identify coefficients**:
- \(2g = -\frac{3}{2} \Rightarrow g = -\frac{3}{4}\)
- \(2f = 1 \Rightarrow f = \frac{1}{2}\)
- \(c = -\frac{1}{2}\)

3. **Calculate center and radius**:
- Center: \((-g, -f) = \left(\frac{3}{4}, -\frac{1}{2}\right)\)
- Radius: \(\sqrt{g^2 + f^2 - c} = \sqrt{\left(-\frac{3}{4}\right)^2 + \left(\frac{1}{2}\right)^2 - (-\frac{1}{2})} = \sqrt{\frac{9}{16} + \frac{1}{4} + \frac{1}{2}} = \sqrt{\frac{9}{16} + \frac{4}{16} + \frac{8}{16}} = \sqrt{\frac{21}{16}} = \frac{\sqrt{21}}{4}\)

### (vi) \(x^2 + y^2 + 2x - 4y - 4 = 0\)

1. **Identify coefficients**:
- \(2g = 2 \Rightarrow g = 1\)
- \(2f = -4 \Rightarrow f = -2\)
- \(c = -4\)

2. **Calculate center and radius**:
- Center: \((-g, -f) = (-1, 2)\)
- Radius: \(\sqrt{g^2 + f^2 - c} = \sqrt{1^2 + (-2)^2 - (-4)} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3\)

### (vii) \(x^2 + y^2 - 4x - 8y - 41 = 0\)

1. **Identify coefficients**:
- \(2g = -4 \Rightarrow g = -2\)
- \(2f = -8 \Rightarrow f = -4\)
- \(c = -41\)

2. **Calculate center and radius**:
- Center: \((-g, -f) = (2, 4)\)
- Radius: \(\sqrt{g^2 + f^2 - c} = \sqrt{(-2)^2 + (-4)^2 - (-41)} = \sqrt{4 + 16 + 41} = \sqrt{61}\)

### Summary of Results:
1. (i) Center: \(\left(\frac{5}{6}, 1\right)\), Radius: \(\frac{\sqrt{13}}{6}\)
2. (ii) Center: \((-1, 2)\), Radius: \(\frac{4}{\sqrt{3}}\)
3. (iii) Center: \((-3, -4)\), Radius: \(11\)
4. (iv) Center: \((1, -\frac{3}{2})\), Radius: \(\frac{\sqrt{19}}{2}\)
5. (v) Center: \(\left(\frac{3}{4}, -\frac{1}{2}\right)\), Radius: \(\frac{\sqrt{21}}{4}\)
6. (vi) Center: \((-1, 2)\), Radius: \(3\)
7. (vii) Center: \((2, 4)\), Radius: \(\sqrt{61}\)