Let a point P be such that its distance from the point (5, 0) is thrice the distance of P from the point (-5, 0). If the locus of the point P is a circle of radius r, then `4r^2` is equal to _______ .

Answer

To solve the problem, we need to find the locus of the point \( P \) such that its distance from the point \( (5, 0) \) is three times its distance from the point \( (-5, 0) \).

### Step-by-Step Solution:

1. **Define Points and Distances**:
Let \( P(h, k) \) be the point we are interested in. The distance from \( P \) to \( A(5, 0) \) is given by:
\[ PA = \sqrt{(h - 5)^2 + (k - 0)^2} = \sqrt{(h - 5)^2 + k^2} \]
The distance from \( P \) to \( B(-5, 0) \) is given by:
\[ PB = \sqrt{(h + 5)^2 + (k - 0)^2} = \sqrt{(h + 5)^2 + k^2} \]

2. **Set Up the Equation**:
According to the problem, we have:
\[ PA = 3 \cdot PB \]
Squaring both sides to eliminate the square roots gives:
\[ (h - 5)^2 + k^2 = 9 \left( (h + 5)^2 + k^2 \right) \]

3. **Expand Both Sides**:
Expanding the left side:
\[ (h - 5)^2 + k^2 = h^2 - 10h + 25 + k^2 \]
Expanding the right side:
\[ 9 \left( (h + 5)^2 + k^2 \right) = 9(h^2 + 10h + 25 + k^2) = 9h^2 + 90h + 225 + 9k^2 \]

4. **Combine and Simplify**:
Setting both sides equal:
\[ h^2 - 10h + 25 + k^2 = 9h^2 + 90h + 225 + 9k^2 \]
Rearranging gives:
\[ h^2 - 9h^2 - 10h - 90h + 25 - 225 + k^2 - 9k^2 = 0 \]
Simplifying further:
\[ -8h^2 - 100h - 8k^2 - 200 = 0 \]
Dividing the entire equation by -8:
\[ h^2 + 12.5h + k^2 + 25 = 0 \]

5. **Complete the Square**:
Completing the square for \( h \):
\[ h^2 + 12.5h = \left(h + \frac{12.5}{2}\right)^2 - \left(\frac{12.5}{2}\right)^2 = \left(h + 6.25\right)^2 - 39.0625 \]
Thus, we have:
\[ \left(h + 6.25\right)^2 + k^2 - 39.0625 + 25 = 0 \]
Which simplifies to:
\[ \left(h + 6.25\right)^2 + k^2 = 14.0625 \]

6. **Identify the Circle**:
The equation represents a circle centered at \( (-6.25, 0) \) with radius \( r = \sqrt{14.0625} \).

7. **Calculate \( 4r^2 \)**:
Since \( r^2 = 14.0625 \):
\[ 4r^2 = 4 \times 14.0625 = 56.25 \]

### Final Answer:
Thus, \( 4r^2 = 56.25 \).

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