If a point P is moving such that the lengths of tangents drawn from P to the circles
`x^(2)+y^(2)-4x-6y-12=0` and
`x^(2)+y^(2)+6x+18y+26=0` are the ratio 2:3, then find the equation to the locus of P.

Answer

To find the equation of the locus of point P, we need to follow these steps:

### Step 1: Identify the circles and their equations
The given circles are:
1. \( x^2 + y^2 - 4x - 6y - 12 = 0 \) (Circle 1)
2. \( x^2 + y^2 + 6x + 18y + 26 = 0 \) (Circle 2)

### Step 2: Rewrite the equations in standard form
We can rewrite the equations of the circles in standard form by completing the square.

**Circle 1:**
\[ x^2 - 4x + y^2 - 6y = 12 \]
Completing the square:
\[ (x-2)^2 - 4 + (y-3)^2 - 9 = 12 \]
\[ (x-2)^2 + (y-3)^2 = 25 \]
This circle has center \( (2, 3) \) and radius \( 5 \).

**Circle 2:**
\[ x^2 + 6x + y^2 + 18y = -26 \]
Completing the square:
\[ (x+3)^2 - 9 + (y+9)^2 - 81 = -26 \]
\[ (x+3)^2 + (y+9)^2 = 64 \]
This circle has center \( (-3, -9) \) and radius \( 8 \).

### Step 3: Length of tangents from point P(h, k)
The lengths of the tangents from point \( P(h, k) \) to the circles can be calculated using the formula:
\[ d = \sqrt{(h - x_0)^2 + (k - y_0)^2 - r^2} \]
where \( (x_0, y_0) \) is the center of the circle and \( r \) is the radius.

**Length of tangent to Circle 1:**
\[ d_1 = \sqrt{(h - 2)^2 + (k - 3)^2 - 25} \]

**Length of tangent to Circle 2:**
\[ d_2 = \sqrt{(h + 3)^2 + (k + 9)^2 - 64} \]

### Step 4: Set up the ratio of the lengths of tangents
According to the problem, the lengths of the tangents are in the ratio \( 2:3 \):
\[ \frac{d_1}{d_2} = \frac{2}{3} \]

### Step 5: Square both sides and cross multiply
Squaring both sides gives:
\[ \frac{d_1^2}{d_2^2} = \frac{4}{9} \]
Cross multiplying results in:
\[ 9d_1^2 = 4d_2^2 \]

### Step 6: Substitute the expressions for \( d_1^2 \) and \( d_2^2 \)
Substituting the expressions we derived:
\[ 9\left((h - 2)^2 + (k - 3)^2 - 25\right) = 4\left((h + 3)^2 + (k + 9)^2 - 64\right) \]

### Step 7: Expand and simplify both sides
Expanding both sides:
Left side:
\[ 9\left((h - 2)^2 + (k - 3)^2 - 25\right) = 9\left(h^2 - 4h + 4 + k^2 - 6k + 9 - 25\right) \]
\[ = 9(h^2 + k^2 - 4h - 6k - 12) \]
\[ = 9h^2 + 9k^2 - 36h - 54k - 108 \]

Right side:
\[ 4\left((h + 3)^2 + (k + 9)^2 - 64\right) = 4\left(h^2 + 6h + 9 + k^2 + 18k + 81 - 64\right) \]
\[ = 4(h^2 + k^2 + 6h + 18k + 26) \]
\[ = 4h^2 + 4k^2 + 24h + 72k + 104 \]

### Step 8: Set both sides equal and simplify
Setting both sides equal:
\[ 9h^2 + 9k^2 - 36h - 54k - 108 = 4h^2 + 4k^2 + 24h + 72k + 104 \]
Rearranging gives:
\[ 5h^2 + 5k^2 - 60h - 126k - 212 = 0 \]

### Final Equation
Thus, the equation of the locus of point P is:
\[ 5h^2 + 5k^2 - 60h - 126k - 212 = 0 \]