How the following pair of circles are situated in the plane ? Also, find the number of common tangents . `(i) x^(2)+(y-1)^(2)=9` and `(x-1)^(2)+y^(2)=25` (ii) `x^(2)+y^(2)-12x-12y=0` and `x^(2)+y^(2)+6x+6y=0`

Explanation

To solve the given problem, we will analyze the two pairs of circles separately and determine their positions relative to each other, as well as the number of common tangents.

### Part (i): Circles Given by the Equations
1. **Circle 1**: \( x^2 + (y - 1)^2 = 9 \)
- Center \( C_1 = (0, 1) \)
- Radius \( r_1 = 3 \) (since \( r^2 = 9 \))

2. **Circle 2**: \( (x - 1)^2 + y^2 = 25 \)
- Center \( C_2 = (1, 0) \)
- Radius \( r_2 = 5 \) (since \( r^2 = 25 \))

#### Step 1: Calculate the Distance Between the Centers
Using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of the centers:
\[ d = \sqrt{(1 - 0)^2 + (0 - 1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

#### Step 2: Compare the Distance with the Radii
Now we compare \( d \) with \( |r_2 - r_1| \):
\[ |r_2 - r_1| = |5 - 3| = 2 \]
Since \( \sqrt{2} < 2 \), this indicates that Circle 2 is completely inside Circle 1 without touching it.

#### Conclusion for Part (i)
- The circles are situated such that Circle 2 is inside Circle 1.
- The number of common tangents is **0** (since they do not touch).

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### Part (ii): Circles Given by the Equations
1. **Circle 1**: \( x^2 + y^2 - 12x - 12y = 0 \)
- Rearranging gives:
\[ (x - 6)^2 + (y - 6)^2 = 72 \]
- Center \( C_1 = (6, 6) \)
- Radius \( r_1 = 6\sqrt{2} \) (since \( r^2 = 72 \))

2. **Circle 2**: \( x^2 + y^2 + 6x + 6y = 0 \)
- Rearranging gives:
\[ (x + 3)^2 + (y + 3)^2 = 18 \]
- Center \( C_2 = (-3, -3) \)
- Radius \( r_2 = 3\sqrt{2} \) (since \( r^2 = 18 \))

#### Step 1: Calculate the Distance Between the Centers
Using the distance formula:
\[ d = \sqrt{(-3 - 6)^2 + (-3 - 6)^2} = \sqrt{(-9)^2 + (-9)^2} = \sqrt{81 + 81} = 9\sqrt{2} \]

#### Step 2: Compare the Distance with the Radii
Now we compare \( d \) with \( r_1 + r_2 \):
\[ r_1 + r_2 = 6\sqrt{2} + 3\sqrt{2} = 9\sqrt{2} \]
Since \( d = r_1 + r_2 \), this indicates that Circle 1 and Circle 2 touch each other externally.

#### Conclusion for Part (ii)
- The circles are situated such that they touch each other externally.
- The number of common tangents is **1** (the tangent at the point of contact).

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### Summary of Results
- For Part (i): Circles are inside each other; number of common tangents = **0**.
- For Part (ii): Circles touch externally; number of common tangents = **1**.