A point moves so that square of its distance from the point `(3,-2)` is numerically equal to its distance from the line `5x-12y=3`. The equation of its locus is ..........

Explanation

To find the equation of the locus of the point \( P(h, k) \) such that the square of its distance from the point \( A(3, -2) \) is equal to its distance from the line \( 5x - 12y = 3 \), we can follow these steps:

### Step 1: Calculate the distance from point A to point P
The distance \( d_1 \) from point \( A(3, -2) \) to point \( P(h, k) \) is given by the formula:
\[ d_1 = \sqrt{(h - 3)^2 + (k + 2)^2} \]
To find the square of this distance, we have:
\[ d_1^2 = (h - 3)^2 + (k + 2)^2 \]

### Step 2: Calculate the distance from point P to the line
The distance \( d_2 \) from point \( P(h, k) \) to the line \( 5x - 12y - 3 = 0 \) is given by the formula:
\[ d_2 = \frac{|5h - 12k - 3|}{\sqrt{5^2 + (-12)^2}} = \frac{|5h - 12k - 3|}{13} \]

### Step 3: Set the distances equal
According to the problem, the square of the distance from point \( A \) to point \( P \) is equal to the distance from point \( P \) to the line:
\[ d_1^2 = d_2 \]
Substituting the expressions we derived:
\[ (h - 3)^2 + (k + 2)^2 = \frac{|5h - 12k - 3|}{13} \]

### Step 4: Remove the absolute value
To eliminate the absolute value, we can consider two cases:
1. \( 5h - 12k - 3 \geq 0 \)
2. \( 5h - 12k - 3 < 0 \)

For simplicity, we will consider the case where \( 5h - 12k - 3 \geq 0 \):
\[ (h - 3)^2 + (k + 2)^2 = \frac{5h - 12k - 3}{13} \]

### Step 5: Multiply through by 13
To eliminate the fraction, multiply both sides by 13:
\[ 13[(h - 3)^2 + (k + 2)^2] = 5h - 12k - 3 \]

### Step 6: Expand and simplify
Expanding the left side:
\[ 13[(h^2 - 6h + 9) + (k^2 + 4k + 4)] = 5h - 12k - 3 \]
This simplifies to:
\[ 13h^2 + 13k^2 - 78h + 52k + 169 = 5h - 12k - 3 \]

### Step 7: Rearrange the equation
Bringing all terms to one side:
\[ 13h^2 + 13k^2 - 78h - 5h + 52k + 12k + 169 + 3 = 0 \]
This simplifies to:
\[ 13h^2 + 13k^2 - 83h + 64k + 172 = 0 \]

### Final Equation
Thus, the equation of the locus of the point \( P(h, k) \) is:
\[ 13x^2 + 13y^2 - 83x + 64y + 172 = 0 \]