Find the locus of a point, so that the join of points (-5, 1) and (3, 2) subtends a right angle at the moving point.

Answer

To find the locus of a point such that the line segment joining the points (-5, 1) and (3, 2) subtends a right angle at the moving point, we can follow these steps:

### Step 1: Identify the Points
Let the points A and B be defined as follows:
- Point A = (-5, 1)
- Point B = (3, 2)

Let the moving point be P(h, k).

### Step 2: Use the Right Angle Condition
For the points A, B, and P to form a right triangle with the right angle at P, we can use the property that the square of the length of the hypotenuse (AB) is equal to the sum of the squares of the lengths of the other two sides (AP and PB).

### Step 3: Calculate the Lengths
Using the distance formula, we can express the lengths as follows:
- Length AB = √[(3 - (-5))^2 + (2 - 1)^2] = √[(3 + 5)^2 + (2 - 1)^2] = √[8^2 + 1^2] = √[64 + 1] = √65
- Length AP = √[(h - (-5))^2 + (k - 1)^2] = √[(h + 5)^2 + (k - 1)^2]
- Length PB = √[(h - 3)^2 + (k - 2)^2]

### Step 4: Apply the Pythagorean Theorem
According to the Pythagorean theorem:
\[ AB^2 = AP^2 + PB^2 \]

Substituting the lengths we calculated:
\[ 65 = (h + 5)^2 + (k - 1)^2 + (h - 3)^2 + (k - 2)^2 \]

### Step 5: Expand and Simplify
Now, expand both sides:
1. Expand \( (h + 5)^2 \):
\[ (h + 5)^2 = h^2 + 10h + 25 \]
2. Expand \( (k - 1)^2 \):
\[ (k - 1)^2 = k^2 - 2k + 1 \]
3. Expand \( (h - 3)^2 \):
\[ (h - 3)^2 = h^2 - 6h + 9 \]
4. Expand \( (k - 2)^2 \):
\[ (k - 2)^2 = k^2 - 4k + 4 \]

Now, substituting these expansions back into the equation:
\[ 65 = (h^2 + 10h + 25) + (k^2 - 2k + 1) + (h^2 - 6h + 9) + (k^2 - 4k + 4) \]

Combine like terms:
\[ 65 = 2h^2 + 2k^2 + (10h - 6h) + (-2k - 4k) + (25 + 1 + 9 + 4) \]
\[ 65 = 2h^2 + 2k^2 + 4h - 6k + 39 \]

### Step 6: Rearranging the Equation
Now, rearranging gives:
\[ 2h^2 + 2k^2 + 4h - 6k + 39 - 65 = 0 \]
\[ 2h^2 + 2k^2 + 4h - 6k - 26 = 0 \]

### Step 7: Divide by 2
To simplify, divide the entire equation by 2:
\[ h^2 + k^2 + 2h - 3k - 13 = 0 \]

### Step 8: Replace h and k with x and y
Since h and k represent the coordinates of the moving point P, we can replace them with x and y:
\[ x^2 + y^2 + 2x - 3y - 13 = 0 \]

### Final Answer
The locus of the point is given by the equation:
\[ x^2 + y^2 + 2x - 3y - 13 = 0 \]