If the locus of the point which moves so that the difference (p) 0 of its distance from the points `(5, 0) and (-5,0)` is 2 is `x^2/a^2-y^2/24=1` then a is

Answer

To solve the problem step by step, we will find the value of \( a \) given the locus of the point \( P(h, k) \) such that the difference of its distances from the points \( (5, 0) \) and \( (-5, 0) \) is 2.

### Step 1: Define the distances
Let \( P(h, k) \) be the point whose locus we are trying to find. The distance from \( P(h, k) \) to \( (5, 0) \) is given by:
\[ d_1 = \sqrt{(h - 5)^2 + k^2} \]
The distance from \( P(h, k) \) to \( (-5, 0) \) is given by:
\[ d_2 = \sqrt{(h + 5)^2 + k^2} \]

### Step 2: Set up the equation
According to the problem, the difference of these distances is 2:
\[ d_1 - d_2 = 2 \]
Substituting the distances we found:
\[ \sqrt{(h - 5)^2 + k^2} - \sqrt{(h + 5)^2 + k^2} = 2 \]

### Step 3: Isolate one of the square roots
Rearranging the equation gives:
\[ \sqrt{(h - 5)^2 + k^2} = 2 + \sqrt{(h + 5)^2 + k^2} \]

### Step 4: Square both sides
Now, we square both sides to eliminate the square root:
\[ (h - 5)^2 + k^2 = (2 + \sqrt{(h + 5)^2 + k^2})^2 \]
Expanding the right side:
\[ (h - 5)^2 + k^2 = 4 + 4\sqrt{(h + 5)^2 + k^2} + (h + 5)^2 + k^2 \]

### Step 5: Simplify the equation
By simplifying, we can cancel \( k^2 \) from both sides:
\[ (h - 5)^2 = 4 + 4\sqrt{(h + 5)^2 + k^2} + (h + 5)^2 \]

### Step 6: Rearranging and isolating the square root
Rearranging gives:
\[ (h - 5)^2 - (h + 5)^2 = 4 + 4\sqrt{(h + 5)^2 + k^2} \]
Using the difference of squares:
\[ [(h - 5) - (h + 5)][(h - 5) + (h + 5)] = 4 + 4\sqrt{(h + 5)^2 + k^2} \]
This simplifies to:
\[ -10 \cdot 2h = 4 + 4\sqrt{(h + 5)^2 + k^2} \]
Thus:
\[ -20h = 4 + 4\sqrt{(h + 5)^2 + k^2} \]

### Step 7: Isolate the square root again
Rearranging gives:
\[ -20h - 4 = 4\sqrt{(h + 5)^2 + k^2} \]
Dividing by 4:
\[ -5h - 1 = \sqrt{(h + 5)^2 + k^2} \]

### Step 8: Square both sides again
Squaring both sides again:
\[ (-5h - 1)^2 = (h + 5)^2 + k^2 \]
Expanding both sides:
\[ 25h^2 + 10h + 1 = h^2 + 10h + 25 + k^2 \]

### Step 9: Rearranging terms
Bringing all terms to one side:
\[ 25h^2 - h^2 + 1 - 25 = k^2 \]
This simplifies to:
\[ 24h^2 - 24 = k^2 \]
Dividing by 24:
\[ \frac{h^2}{1} - \frac{k^2}{24} = 1 \]

### Step 10: Identify the value of \( a \)
From the equation \( \frac{x^2}{a^2} - \frac{y^2}{24} = 1 \), we can see that \( a^2 = 1 \), hence:
\[ a = \pm 1 \]

### Final Answer
Thus, the value of \( a \) is \( 1 \).

---