Find the equation of circle whose :
(i) radius is 5 and centre is (3,4).
(ii) radius is `sqrt(5)` and centre is (0.2).
(iii) radius is `sqrt(a^(2)+b^(2))` and centre is (a,b).
(iv) radius is r and centre is `(rcostheta,rsintheta)`.
(v) radius is `sqrt(a^(2)sec^(2)theta+b^(2)tan^(2)theta)` and centre is `(asectheta,btantheta)`.

Answer

To find the equations of the circles based on the given parameters, we will use the standard equation of a circle, which is:

\[ (x - a)^2 + (y - b)^2 = r^2 \]

where \((a, b)\) is the center of the circle and \(r\) is the radius.

### (i) Radius is 5 and center is (3, 4)

1. **Identify the center and radius**:
- Center \((a, b) = (3, 4)\)
- Radius \(r = 5\)

2. **Substitute into the equation**:
\[ (x - 3)^2 + (y - 4)^2 = 5^2 \]

3. **Simplify**:
\[ (x - 3)^2 + (y - 4)^2 = 25 \]

4. **Expand the equation**:
\[ x^2 - 6x + 9 + y^2 - 8y + 16 = 25 \]

5. **Combine like terms**:
\[ x^2 + y^2 - 6x - 8y + 25 - 25 = 0 \]
\[ x^2 + y^2 - 6x - 8y = 0 \]

### (ii) Radius is \(\sqrt{5}\) and center is (0, 2)

1. **Identify the center and radius**:
- Center \((a, b) = (0, 2)\)
- Radius \(r = \sqrt{5}\)

2. **Substitute into the equation**:
\[ (x - 0)^2 + (y - 2)^2 = (\sqrt{5})^2 \]

3. **Simplify**:
\[ x^2 + (y - 2)^2 = 5 \]

4. **Expand the equation**:
\[ x^2 + (y^2 - 4y + 4) = 5 \]

5. **Combine like terms**:
\[ x^2 + y^2 - 4y + 4 - 5 = 0 \]
\[ x^2 + y^2 - 4y - 1 = 0 \]

### (iii) Radius is \(\sqrt{a^2 + b^2}\) and center is (a, b)

1. **Identify the center and radius**:
- Center \((a, b) = (a, b)\)
- Radius \(r = \sqrt{a^2 + b^2}\)

2. **Substitute into the equation**:
\[ (x - a)^2 + (y - b)^2 = (\sqrt{a^2 + b^2})^2 \]

3. **Simplify**:
\[ (x - a)^2 + (y - b)^2 = a^2 + b^2 \]

4. **Expand the equation**:
\[ x^2 - 2ax + a^2 + y^2 - 2by + b^2 = a^2 + b^2 \]

5. **Combine like terms**:
\[ x^2 + y^2 - 2ax - 2by + a^2 + b^2 - a^2 - b^2 = 0 \]
\[ x^2 + y^2 - 2ax - 2by = 0 \]

### (iv) Radius is \(r\) and center is \((r \cos \theta, r \sin \theta)\)

1. **Identify the center and radius**:
- Center \((a, b) = (r \cos \theta, r \sin \theta)\)
- Radius \(r = r\)

2. **Substitute into the equation**:
\[ (x - r \cos \theta)^2 + (y - r \sin \theta)^2 = r^2 \]

3. **Expand the equation**:
\[ (x^2 - 2xr \cos \theta + r^2 \cos^2 \theta) + (y^2 - 2yr \sin \theta + r^2 \sin^2 \theta) = r^2 \]

4. **Combine like terms**:
\[ x^2 + y^2 - 2xr \cos \theta - 2yr \sin \theta + r^2 (\cos^2 \theta + \sin^2 \theta) = r^2 \]

5. **Use the identity \(\cos^2 \theta + \sin^2 \theta = 1\)**:
\[ x^2 + y^2 - 2xr \cos \theta - 2yr \sin \theta + r^2 = r^2 \]

6. **Simplify**:
\[ x^2 + y^2 - 2xr \cos \theta - 2yr \sin \theta = 0 \]

### (v) Radius is \(\sqrt{a^2 \sec^2 \theta + b^2 \tan^2 \theta}\) and center is \((a \sec \theta, b \tan \theta)\)

1. **Identify the center and radius**:
- Center \((a, b) = (a \sec \theta, b \tan \theta)\)
- Radius \(r = \sqrt{a^2 \sec^2 \theta + b^2 \tan^2 \theta}\)

2. **Substitute into the equation**:
\[ (x - a \sec \theta)^2 + (y - b \tan \theta)^2 = (\sqrt{a^2 \sec^2 \theta + b^2 \tan^2 \theta})^2 \]

3. **Simplify**:
\[ (x - a \sec \theta)^2 + (y - b \tan \theta)^2 = a^2 \sec^2 \theta + b^2 \tan^2 \theta \]

4. **Expand the equation**:
\[ (x^2 - 2x a \sec \theta + a^2 \sec^2 \theta) + (y^2 - 2y b \tan \theta + b^2 \tan^2 \theta) = a^2 \sec^2 \theta + b^2 \tan^2 \theta \]

5. **Combine like terms**:
\[ x^2 + y^2 - 2x a \sec \theta - 2y b \tan \theta + a^2 \sec^2 \theta + b^2 \tan^2 \theta - a^2 \sec^2 \theta - b^2 \tan^2 \theta = 0 \]

6. **Simplify**:
\[ x^2 + y^2 - 2x a \sec \theta - 2y b \tan \theta = 0 \]