Find the equation to the pair of tangents drawn
from the point (11,3) to the circle ` x^(2) + y^(2) = 65`

Answer

To find the equation of the pair of tangents drawn from the point (11, 3) to the circle given by the equation \(x^2 + y^2 = 65\), we can follow these steps:

### Step 1: Write the equation of the circle in standard form
The given circle is already in standard form:
\[ x^2 + y^2 - 65 = 0 \]
Here, \(g = 0\), \(f = 0\), and \(c = -65\).

### Step 2: Identify the point from which the tangents are drawn
The point from which the tangents are drawn is \(P(11, 3)\). We denote this point as \((x_1, y_1) = (11, 3)\).

### Step 3: Use the formula for the pair of tangents
The equation of the pair of tangents from a point \((x_1, y_1)\) to the circle \(x^2 + y^2 + 2gx + 2fy + c = 0\) is given by:
\[ S \cdot S_1 = T^2 \]
Where:
- \(S = x^2 + y^2 + 2gx + 2fy + c\)
- \(S_1 = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c\)
- \(T = xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c\)

### Step 4: Calculate \(S\) and \(S_1\)
For our circle:
\[ S = x^2 + y^2 + 0 + 0 - 65 = x^2 + y^2 - 65 \]

Now, calculate \(S_1\):
\[ S_1 = 11^2 + 3^2 + 0 + 0 - 65 = 121 + 9 - 65 = 65 \]

### Step 5: Write the equation for \(T\)
The equation for \(T\) is:
\[ T = xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c \]
Substituting the values:
\[ T = 11x + 3y - 65 \]

### Step 6: Substitute \(S\), \(S_1\), and \(T\) into the equation
Now substitute \(S\), \(S_1\), and \(T\) into the equation \(S \cdot S_1 = T^2\):
\[ (x^2 + y^2 - 65) \cdot 65 = (11x + 3y - 65)^2 \]

### Step 7: Expand both sides
Expanding the left side:
\[ 65x^2 + 65y^2 - 4225 = (11x + 3y - 65)^2 \]

Expanding the right side:
\[ (11x + 3y - 65)^2 = 121x^2 + 9y^2 + 2 \cdot 11 \cdot 3xy - 2 \cdot 65 \cdot 11x - 2 \cdot 65 \cdot 3y + 65^2 \]
\[ = 121x^2 + 9y^2 + 66xy - 1430x - 390y + 4225 \]

### Step 8: Set the equation to zero
Now, we equate both sides:
\[ 65x^2 + 65y^2 - 4225 = 121x^2 + 9y^2 + 66xy - 1430x - 390y + 4225 \]

Rearranging gives:
\[ 65x^2 - 121x^2 + 65y^2 - 9y^2 - 66xy + 1430x + 390y - 4225 - 4225 = 0 \]
\[ -56x^2 + 56y^2 - 66xy + 1430x + 390y - 8450 = 0 \]

### Step 9: Simplify the equation
Dividing through by -1 gives:
\[ 56x^2 - 56y^2 + 66xy - 1430x - 390y + 8450 = 0 \]

### Final Equation
The final equation of the pair of tangents is:
\[ 28x^2 - 28y^2 + 33xy - 715x - 195y + 4225 = 0 \]