If the distance from 'P' to the points (2,3) and (2,-3) are in the ratio 2:3, then find the equation of the locus of P.

Answer

To find the locus of point P such that the distances from P to the points (2, 3) and (2, -3) are in the ratio 2:3, we can follow these steps:

### Step 1: Define the points and the distances
Let the point P be represented as (x, y). The points A and B are given as A(2, 3) and B(2, -3). We need to find the distances PA and PB.

### Step 2: Write the distance formulas
Using the distance formula, we can express PA and PB as:
- \( PA = \sqrt{(x - 2)^2 + (y - 3)^2} \)
- \( PB = \sqrt{(x - 2)^2 + (y + 3)^2} \)

### Step 3: Set up the ratio of distances
According to the problem, the ratio of the distances PA to PB is given as:
\[ \frac{PA}{PB} = \frac{2}{3} \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 3 \cdot PA = 2 \cdot PB \]

### Step 5: Substitute the distance formulas
Substituting the expressions for PA and PB:
\[ 3 \sqrt{(x - 2)^2 + (y - 3)^2} = 2 \sqrt{(x - 2)^2 + (y + 3)^2} \]

### Step 6: Square both sides to eliminate the square roots
Squaring both sides results in:
\[ 9 \left((x - 2)^2 + (y - 3)^2\right) = 4 \left((x - 2)^2 + (y + 3)^2\right) \]

### Step 7: Expand both sides
Expanding both sides:
- Left side:
\[ 9 \left((x - 2)^2 + (y - 3)^2\right) = 9 \left((x - 2)^2 + (y^2 - 6y + 9)\right) = 9(x^2 - 4x + 4 + y^2 - 6y + 9) = 9x^2 + 9y^2 - 36y - 36x + 117 \]
- Right side:
\[ 4 \left((x - 2)^2 + (y + 3)^2\right) = 4 \left((x - 2)^2 + (y^2 + 6y + 9)\right) = 4(x^2 - 4x + 4 + y^2 + 6y + 9) = 4x^2 + 4y^2 + 24y - 16x + 52 \]

### Step 8: Set the expanded forms equal to each other
Setting both expanded forms equal gives:
\[ 9x^2 + 9y^2 - 36y - 36x + 117 = 4x^2 + 4y^2 + 24y - 16x + 52 \]

### Step 9: Move all terms to one side
Rearranging the equation leads to:
\[ 5x^2 + 5y^2 - 20x - 60y + 65 = 0 \]

### Step 10: Simplify the equation
Dividing the entire equation by 5 gives the final equation of the locus:
\[ x^2 + y^2 - 4x - 12y + 13 = 0 \]

### Final Answer:
The equation of the locus of point P is:
\[ 5x^2 + 5y^2 - 20x - 78y + 65 = 0 \]