Find the locus of the point, the sum of whose distances from the points `A(4,0,0)a n d\ B(-4,0,0)` is equal to 10.

Explanation

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

### Step 1: Define the Points and the Locus
Let \( P(x, y, z) \) be the point whose distances from points \( A \) and \( B \) we are considering. We know that the sum of distances from \( P \) to \( A \) and \( B \) is equal to 10:
\[ PA + PB = 10 \]

### Step 2: Apply the Distance Formula
Using the distance formula, we can express \( PA \) and \( PB \):
\[ PA = \sqrt{(x - 4)^2 + y^2 + z^2} \]
\[ PB = \sqrt{(x + 4)^2 + y^2 + z^2} \]
Thus, we can write:
\[ \sqrt{(x - 4)^2 + y^2 + z^2} + \sqrt{(x + 4)^2 + y^2 + z^2} = 10 \]

### Step 3: Square Both Sides
To eliminate the square roots, we can square both sides of the equation. However, we will first isolate one of the square root terms:
\[ \sqrt{(x - 4)^2 + y^2 + z^2} = 10 - \sqrt{(x + 4)^2 + y^2 + z^2} \]
Now, squaring both sides gives:
\[ (x - 4)^2 + y^2 + z^2 = (10 - \sqrt{(x + 4)^2 + y^2 + z^2})^2 \]

### Step 4: Expand and Simplify
Expanding the right side:
\[ (x - 4)^2 + y^2 + z^2 = 100 - 20\sqrt{(x + 4)^2 + y^2 + z^2} + (x + 4)^2 + y^2 + z^2 \]
Now, simplify:
\[ (x - 4)^2 = 100 - 20\sqrt{(x + 4)^2 + y^2 + z^2} + (x + 4)^2 \]

### Step 5: Rearranging the Equation
Rearranging gives:
\[ (x - 4)^2 - (x + 4)^2 = 100 - 20\sqrt{(x + 4)^2 + y^2 + z^2} \]
Using the difference of squares:
\[ [(x - 4) - (x + 4)][(x - 4) + (x + 4)] = 100 - 20\sqrt{(x + 4)^2 + y^2 + z^2} \]
This simplifies to:
\[ -8(2x) = 100 - 20\sqrt{(x + 4)^2 + y^2 + z^2} \]
\[ -16x = 100 - 20\sqrt{(x + 4)^2 + y^2 + z^2} \]

### Step 6: Isolate the Square Root
Isolating the square root gives:
\[ 20\sqrt{(x + 4)^2 + y^2 + z^2} = 100 + 16x \]
Now divide by 20:
\[ \sqrt{(x + 4)^2 + y^2 + z^2} = 5 + \frac{4x}{5} \]

### Step 7: Square Again
Squaring both sides again:
\[ (x + 4)^2 + y^2 + z^2 = \left(5 + \frac{4x}{5}\right)^2 \]
Expanding the right-hand side:
\[ (x + 4)^2 + y^2 + z^2 = 25 + 8x + \frac{16x^2}{25} \]

### Step 8: Collect Terms
Now, collect all terms:
\[ x^2 + 8x + 16 + y^2 + z^2 = 25 + 8x + \frac{16x^2}{25} \]
Rearranging gives:
\[ x^2 + y^2 + z^2 = 9 + \frac{16x^2}{25} \]

### Step 9: Final Form
This can be simplified to find the equation of the locus. After further simplification, we arrive at the final equation:
\[ 21x^2 + 25y^2 + 25z^2 = 525 \]

### Conclusion
The locus of the point is an ellipsoid given by the equation:
\[ 21x^2 + 25y^2 + 25z^2 = 525 \]