Find the equation of the locus of point P, the sum of whose distance from the points (2, 0, 0)
and (-4, 0, 0) is 10.

Answer

To find the equation of the locus of point P, whose sum of distances from the points A(2, 0, 0) and B(-4, 0, 0) is 10, we can follow these steps:

### Step 1: Define the points and distances
Let point P be represented as \( P(x, y, z) \). The distances from point P to points A and B can be expressed as:
- Distance from P to A: \( AP = \sqrt{(x - 2)^2 + y^2 + z^2} \)
- Distance from P to B: \( PB = \sqrt{(x + 4)^2 + y^2 + z^2} \)

### Step 2: Set up the equation based on the given condition
According to the problem, the sum of these distances is equal to 10:
\[ AP + PB = 10 \]
This gives us the equation:
\[ \sqrt{(x - 2)^2 + y^2 + z^2} + \sqrt{(x + 4)^2 + y^2 + z^2} = 10 \]

### Step 3: Isolate one of the square root terms
To simplify, we can isolate one of the square root terms:
\[ \sqrt{(x - 2)^2 + y^2 + z^2} = 10 - \sqrt{(x + 4)^2 + y^2 + z^2} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides, we get:
\[ (x - 2)^2 + y^2 + z^2 = (10 - \sqrt{(x + 4)^2 + y^2 + z^2})^2 \]
Expanding the right side:
\[ (x - 2)^2 + y^2 + z^2 = 100 - 20\sqrt{(x + 4)^2 + y^2 + z^2} + (x + 4)^2 + y^2 + z^2 \]

### Step 5: Simplify the equation
Now, we can simplify this equation:
\[ (x - 2)^2 = 100 - 20\sqrt{(x + 4)^2 + y^2 + z^2} + (x + 4)^2 \]
This leads to:
\[ (x - 2)^2 - (x + 4)^2 = 100 - 20\sqrt{(x + 4)^2 + y^2 + z^2} \]

### Step 6: Further simplify and isolate the square root
Now, we will expand both squares and simplify:
\[ (x^2 - 4x + 4) - (x^2 + 8x + 16) = 100 - 20\sqrt{(x + 4)^2 + y^2 + z^2} \]
This simplifies to:
\[ -12x - 12 = 100 - 20\sqrt{(x + 4)^2 + y^2 + z^2} \]

### Step 7: Isolate the square root again and square both sides
Rearranging gives:
\[ 20\sqrt{(x + 4)^2 + y^2 + z^2} = 112 + 12x \]
Now squaring both sides again:
\[ 400((x + 4)^2 + y^2 + z^2) = (112 + 12x)^2 \]

### Step 8: Expand and simplify
Expanding both sides leads to:
\[ 400(x^2 + 8x + 16 + y^2 + z^2) = 12544 + 2688x + 144x^2 \]
This simplifies to:
\[ 256x^2 + 512x + 400y^2 + 400z^2 - 6144 = 0 \]

### Step 9: Final equation
Dividing through by 16 gives:
\[ 16x^2 + 25y^2 + 25z^2 + 32x - 384 = 0 \]

Thus, the final equation of the locus of point P is:
\[ 16x^2 + 25y^2 + 25z^2 + 32x - 384 = 0 \]