Find the equation to the pair of tangents drawn
From the point (4,5) to the circle
` 2 x ^(2) + 2y^(2) - 8 x + 12 y + 21 = 0`

Answer

To find the equation of the pair of tangents drawn from the point (4, 5) to the circle given by the equation \(2x^2 + 2y^2 - 8x + 12y + 21 = 0\), we will follow these steps:

### Step 1: Rewrite the circle equation in standard form
First, we simplify the given circle equation by dividing all terms by 2:

\[ x^2 + y^2 - 4x + 6y + \frac{21}{2} = 0 \]

### Step 2: Identify the coefficients
From the standard form of the circle equation \(x^2 + y^2 + 2gx + 2fy + c = 0\), we identify:
- \(g = -4\)
- \(f = 3\)
- \(c = \frac{21}{2}\)

### Step 3: Calculate \(S\) and \(S_1\)
The general point \(P(x_1, y_1)\) is given as (4, 5). We calculate \(S\) and \(S_1\):

\[ S = x^2 + y^2 + 2gx + 2fy + c \]
Substituting \(g\), \(f\), and \(c\):

\[ S = 4^2 + 5^2 + 2(-4)(4) + 2(3)(5) + \frac{21}{2} \]
Calculating each term:
- \(4^2 = 16\)
- \(5^2 = 25\)
- \(2(-4)(4) = -32\)
- \(2(3)(5) = 30\)

Now substituting these values:

\[ S = 16 + 25 - 32 + 30 + \frac{21}{2} \]
Calculating \(S\):

\[ S = 39 - 32 + \frac{21}{2} = 7 + \frac{21}{2} = \frac{14}{2} + \frac{21}{2} = \frac{35}{2} \]

Now, calculate \(S_1\):

\[ S_1 = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c \]
Substituting \(x_1 = 4\) and \(y_1 = 5\):

\[ S_1 = 4^2 + 5^2 + 2(-4)(4) + 2(3)(5) + \frac{21}{2} \]
This is the same calculation as above, so:

\[ S_1 = \frac{35}{2} \]

### Step 4: Use the formula \(S \cdot S_1 = T^2\)
The equation of the tangents can be found using the relation:

\[ S \cdot S_1 = T^2 \]

Where \(T\) is given by:

\[ T = x^2 + y^2 + 2gx_1x + 2fy_1y + c \]

Substituting \(x_1 = 4\) and \(y_1 = 5\):

\[ T = x^2 + y^2 + 2(-4)(x) + 2(3)(y) + \frac{21}{2} \]

### Step 5: Substitute \(S\) and \(S_1\) into the equation
Now substituting:

\[ \frac{35}{2} \cdot \frac{35}{2} = T^2 \]

Calculating:

\[ \frac{1225}{4} = T^2 \]

### Step 6: Expand \(T\)
Expanding \(T\):

\[ T = x^2 + y^2 - 8x + 6y + \frac{21}{2} \]

### Step 7: Set up the equation
Setting up the equation:

\[ \frac{1225}{4} = \left(x^2 + y^2 - 8x + 6y + \frac{21}{2}\right)^2 \]

### Step 8: Rearranging
Rearranging gives us the equation of the pair of tangents:

\[ 4(x^2 + y^2 - 8x + 6y + \frac{21}{2})^2 - 1225 = 0 \]

### Final Equation
This results in the final equation of the pair of tangents from the point (4, 5) to the circle.