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 locus of a point that moves such that the square of its distance from the point (3, -2) is numerically equal to its distance from the line \(5x - 12y = 3\), we can follow these steps:

### Step 1: Define the point and distances
Let the moving point be \(P(h, k)\). The distance \(d_1\) from point \(P\) to the point \((3, -2)\) is given by the distance formula:
\[ d_1 = \sqrt{(h - 3)^2 + (k + 2)^2} \]
The square of this distance is:
\[ d_1^2 = (h - 3)^2 + (k + 2)^2 \]

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

### Step 3: Set up the equation
According to the problem, the square of the distance from the point to (3, -2) is equal to the distance from the line:
\[ (h - 3)^2 + (k + 2)^2 = \left(\frac{|5h - 12k - 3|}{13}\right)^2 \]

### Step 4: Eliminate the absolute value
We can square both sides to eliminate the square root:
\[ (h - 3)^2 + (k + 2)^2 = \frac{(5h - 12k - 3)^2}{169} \]

### Step 5: Multiply through by 169
To eliminate the fraction, multiply both sides by 169:
\[ 169[(h - 3)^2 + (k + 2)^2] = (5h - 12k - 3)^2 \]

### Step 6: Expand both sides
Now, expand both sides:
- Left side:
\[ 169[(h - 3)^2 + (k + 2)^2] = 169[(h^2 - 6h + 9) + (k^2 + 4k + 4)] = 169h^2 - 1014h + 169 \cdot 13 + 169k^2 + 676k \]
- Right side:
\[ (5h - 12k - 3)^2 = 25h^2 - 120hk - 30h + 144k^2 + 72k + 9 \]

### Step 7: Rearranging the equation
After expanding, we can rearrange the equation to bring all terms to one side:
\[ 169h^2 + 169k^2 - 83h + 64k + 172 = 0 \]

### Step 8: Replace variables
Finally, replace \(h\) and \(k\) with \(x\) and \(y\):
\[ 13x^2 + 13y^2 - 83x + 64y + 172 = 0 \]

### Final Answer
Thus, the equation of the locus is:
\[ 13x^2 + 13y^2 - 83x + 64y + 172 = 0 \]