Find the equation of the locus of a point which moves so that the difference of its distances from the points (3, 0) and (-3, 0) is 4 units.

Answer

To find the equation of the locus of a point that moves such that the difference of its distances from the points (3, 0) and (-3, 0) is 4 units, we can follow these steps:

### Step 1: Define the point and distances
Let the moving point be \( P(h, k) \). The distances from this point to the points (3, 0) and (-3, 0) can be expressed as:
- Distance from \( P(h, k) \) to \( (3, 0) \):
\[ d_1 = \sqrt{(h - 3)^2 + (k - 0)^2} = \sqrt{(h - 3)^2 + k^2} \]
- Distance from \( P(h, k) \) to \( (-3, 0) \):
\[ d_2 = \sqrt{(h + 3)^2 + (k - 0)^2} = \sqrt{(h + 3)^2 + k^2} \]

### Step 2: Set up the equation based on the condition
According to the problem, the difference of these distances is 4 units:
\[ |d_1 - d_2| = 4 \]
This gives us two cases:
1. \( d_1 - d_2 = 4 \)
2. \( d_2 - d_1 = 4 \)

We will solve the first case:
\[ \sqrt{(h - 3)^2 + k^2} - \sqrt{(h + 3)^2 + k^2} = 4 \]

### Step 3: Square both sides
To eliminate the square roots, we will square both sides:
\[ \left(\sqrt{(h - 3)^2 + k^2} - \sqrt{(h + 3)^2 + k^2}\right)^2 = 16 \]
Expanding the left side:
\[ (h - 3)^2 + k^2 + (h + 3)^2 + k^2 - 2\sqrt{((h - 3)^2 + k^2)((h + 3)^2 + k^2)} = 16 \]

### Step 4: Simplify the equation
Combine like terms:
\[ (h - 3)^2 + (h + 3)^2 + 2k^2 - 16 = 2\sqrt{((h - 3)^2 + k^2)((h + 3)^2 + k^2)} \]
Calculating \( (h - 3)^2 + (h + 3)^2 \):
\[ = (h^2 - 6h + 9) + (h^2 + 6h + 9) = 2h^2 + 18 \]
Thus, we have:
\[ 2h^2 + 2k^2 + 18 - 16 = 2\sqrt{((h - 3)^2 + k^2)((h + 3)^2 + k^2)} \]
This simplifies to:
\[ 2h^2 + 2k^2 + 2 = 2\sqrt{((h - 3)^2 + k^2)((h + 3)^2 + k^2)} \]
Dividing everything by 2:
\[ h^2 + k^2 + 1 = \sqrt{((h - 3)^2 + k^2)((h + 3)^2 + k^2)} \]

### Step 5: Square both sides again
Now we square both sides again:
\[ (h^2 + k^2 + 1)^2 = ((h - 3)^2 + k^2)((h + 3)^2 + k^2) \]
Expanding both sides:
Left side:
\[ h^4 + 2h^2k^2 + k^4 + 2h^2 + 2k^2 + 1 \]
Right side:
\[ ((h^2 - 6h + 9) + k^2)((h^2 + 6h + 9) + k^2) \]
This results in:
\[ (h^2 + k^2)^2 + 18 + 2k^2 - 36h^2 = 0 \]

### Step 6: Rearranging the equation
After simplification, we find:
\[ 5h^2 - 4k^2 = 16 \]
This is the equation of the locus.

### Final Result
Thus, the equation of the locus of the point is:
\[ 5h^2 - 4k^2 = 16 \]