Find the equation of the circle with centre C and radius r where
(i) `C(-1,b),r=a+b`
(ii) `C=(-a,-b),r=sqrt(a^(2)-b^(2))(|a|gt|b|)`
(iii) `C=(cos theta, sin theta),r=1`
(iv) `C=(-7,-3),r=4`
(v) `C-(-1/2,-9),r=5`
(vi) `C=(1,7),r=5/2`
(vii) `C=(0,0),r=8`
(vii) `C=(2,-3),r=4`
(ix) `C=(-1,2),r=5`

Answer

To find the equation of the circle with center \( C \) and radius \( r \), we use the standard form of the equation of a circle, which is:

\[ (x - h)^2 + (y - k)^2 = r^2 \]

where \( (h, k) \) is the center of the circle and \( r \) is the radius.

Let's solve each part step by step.

### (i) \( C(-1, b), r = a + b \)

1. Identify \( h = -1 \), \( k = b \), and \( r = a + b \).
2. Substitute into the equation:
\[ (x - (-1))^2 + (y - b)^2 = (a + b)^2 \]
3. Simplify:
\[ (x + 1)^2 + (y - b)^2 = (a + b)^2 \]

### (ii) \( C=(-a,-b), r=\sqrt{a^2-b^2} \) where \( |a| > |b| \)

1. Identify \( h = -a \), \( k = -b \), and \( r = \sqrt{a^2 - b^2} \).
2. Substitute into the equation:
\[ (x - (-a))^2 + (y - (-b))^2 = (\sqrt{a^2 - b^2})^2 \]
3. Simplify:
\[ (x + a)^2 + (y + b)^2 = a^2 - b^2 \]

### (iii) \( C=(\cos \theta, \sin \theta), r=1 \)

1. Identify \( h = \cos \theta \), \( k = \sin \theta \), and \( r = 1 \).
2. Substitute into the equation:
\[ (x - \cos \theta)^2 + (y - \sin \theta)^2 = 1^2 \]
3. Simplify:
\[ (x - \cos \theta)^2 + (y - \sin \theta)^2 = 1 \]

### (iv) \( C=(-7,-3), r=4 \)

1. Identify \( h = -7 \), \( k = -3 \), and \( r = 4 \).
2. Substitute into the equation:
\[ (x - (-7))^2 + (y - (-3))^2 = 4^2 \]
3. Simplify:
\[ (x + 7)^2 + (y + 3)^2 = 16 \]

### (v) \( C=(-\frac{1}{2},-9), r=5 \)

1. Identify \( h = -\frac{1}{2} \), \( k = -9 \), and \( r = 5 \).
2. Substitute into the equation:
\[ (x - (-\frac{1}{2}))^2 + (y - (-9))^2 = 5^2 \]
3. Simplify:
\[ (x + \frac{1}{2})^2 + (y + 9)^2 = 25 \]

### (vi) \( C=(1,7), r=\frac{5}{2} \)

1. Identify \( h = 1 \), \( k = 7 \), and \( r = \frac{5}{2} \).
2. Substitute into the equation:
\[ (x - 1)^2 + (y - 7)^2 = \left(\frac{5}{2}\right)^2 \]
3. Simplify:
\[ (x - 1)^2 + (y - 7)^2 = \frac{25}{4} \]

### (vii) \( C=(0,0), r=8 \)

1. Identify \( h = 0 \), \( k = 0 \), and \( r = 8 \).
2. Substitute into the equation:
\[ (x - 0)^2 + (y - 0)^2 = 8^2 \]
3. Simplify:
\[ x^2 + y^2 = 64 \]

### (viii) \( C=(2,-3), r=4 \)

1. Identify \( h = 2 \), \( k = -3 \), and \( r = 4 \).
2. Substitute into the equation:
\[ (x - 2)^2 + (y - (-3))^2 = 4^2 \]
3. Simplify:
\[ (x - 2)^2 + (y + 3)^2 = 16 \]

### (ix) \( C=(-1,2), r=5 \)

1. Identify \( h = -1 \), \( k = 2 \), and \( r = 5 \).
2. Substitute into the equation:
\[ (x - (-1))^2 + (y - 2)^2 = 5^2 \]
3. Simplify:
\[ (x + 1)^2 + (y - 2)^2 = 25 \]