Show that x = 7 and y = 8 touch the circle `x^(2) + y^(2) - 4x - 6y -1 2 = 0` and find the points of contact.

Answer

To solve the problem, we need to show that the lines \(x = 7\) and \(y = 8\) touch the circle given by the equation \(x^2 + y^2 - 4x - 6y - 12 = 0\), and then find the points of contact.

### Step 1: Rewrite the Circle Equation
We start with the equation of the circle:
\[ x^2 + y^2 - 4x - 6y - 12 = 0 \]
We can rearrange this equation to find the center and radius of the circle.

### Step 2: Complete the Square
To complete the square for \(x\) and \(y\):
1. For \(x\):
\[ x^2 - 4x = (x - 2)^2 - 4 \]
2. For \(y\):
\[ y^2 - 6y = (y - 3)^2 - 9 \]

Now substituting these back into the circle equation:
\[ (x - 2)^2 - 4 + (y - 3)^2 - 9 - 12 = 0 \]
This simplifies to:
\[ (x - 2)^2 + (y - 3)^2 - 25 = 0 \]
Thus, we can rewrite it as:
\[ (x - 2)^2 + (y - 3)^2 = 25 \]

### Step 3: Identify the Center and Radius
From the equation \((x - 2)^2 + (y - 3)^2 = 25\), we can identify:
- Center \(C(2, 3)\)
- Radius \(r = 5\)

### Step 4: Check if \(x = 7\) is a Tangent
To check if the line \(x = 7\) is a tangent, we find the distance from the center of the circle to the line \(x = 7\):
- The distance from point \(C(2, 3)\) to the line \(x = 7\) is:
\[ |7 - 2| = 5 \]
Since this distance equals the radius of the circle, \(x = 7\) is indeed a tangent.

### Step 5: Find the Point of Contact for \(x = 7\)
To find the point of contact, substitute \(x = 7\) into the circle equation:
\[ (7 - 2)^2 + (y - 3)^2 = 25 \]
This simplifies to:
\[ 5^2 + (y - 3)^2 = 25 \]
\[ 25 + (y - 3)^2 = 25 \]
\[ (y - 3)^2 = 0 \]
Thus, \(y - 3 = 0\) or \(y = 3\). Therefore, the point of contact is:
\[ (7, 3) \]

### Step 6: Check if \(y = 8\) is a Tangent
Now, we check if the line \(y = 8\) is a tangent:
- The distance from the center \(C(2, 3)\) to the line \(y = 8\) is:
\[ |8 - 3| = 5 \]
Since this distance also equals the radius of the circle, \(y = 8\) is a tangent.

### Step 7: Find the Point of Contact for \(y = 8\)
To find the point of contact, substitute \(y = 8\) into the circle equation:
\[ (x - 2)^2 + (8 - 3)^2 = 25 \]
This simplifies to:
\[ (x - 2)^2 + 5^2 = 25 \]
\[ (x - 2)^2 + 25 = 25 \]
\[ (x - 2)^2 = 0 \]
Thus, \(x - 2 = 0\) or \(x = 2\). Therefore, the point of contact is:
\[ (2, 8) \]

### Final Points of Contact
The points of contact are:
1. \( (7, 3) \) for the line \(x = 7\)
2. \( (2, 8) \) for the line \(y = 8\)

### Summary
Thus, we have shown that \(x = 7\) and \(y = 8\) touch the circle, and the points of contact are \((7, 3)\) and \((2, 8)\).

---