Let `P-=(a, 0), Q-=(-1, 0) and R-=(2, 0)` are three given points. If the locus of the point S satisfying the reaction `SQ^(2)+SR^(2)=2SP^(2)` is `2x+3=0`. Then the sum of all possible values of a is

Options

A. 1
B. `-4`
C. 3
D. `-3`

Answer

To solve the problem, we need to find the values of \( a \) such that the locus of point \( S \) satisfies the equation \( SQ^2 + SR^2 = 2SP^2 \) and is given by the line \( 2x + 3 = 0 \).

### Step-by-step Solution:

1. **Identify the Points**:
- Let \( P = (a, 0) \)
- \( Q = (-1, 0) \)
- \( R = (2, 0) \)

2. **Assume Coordinates of Point \( S \)**:
- Let \( S = (h, k) \)

3. **Write the Distance Formulas**:
- The distances are:
- \( SQ^2 = (h + 1)^2 + k^2 \)
- \( SR^2 = (h - 2)^2 + k^2 \)
- \( SP^2 = (h - a)^2 + k^2 \)

4. **Set Up the Equation**:
- According to the problem, we have:
\[ SQ^2 + SR^2 = 2SP^2 \]
- Substituting the distances:
\[ (h + 1)^2 + k^2 + (h - 2)^2 + k^2 = 2((h - a)^2 + k^2) \]

5. **Simplify the Equation**:
- Combine the left-hand side:
\[ (h + 1)^2 + (h - 2)^2 + 2k^2 = 2((h - a)^2 + k^2) \]
- Expanding both sides:
\[ (h^2 + 2h + 1) + (h^2 - 4h + 4) + 2k^2 = 2(h^2 - 2ah + a^2 + k^2) \]
- Combine like terms:
\[ 2h^2 - 2h + 5 + 2k^2 = 2h^2 - 4ah + 2a^2 + 2k^2 \]

6. **Eliminate Common Terms**:
- Cancel \( 2h^2 \) and \( 2k^2 \) from both sides:
\[ -2h + 5 = -4ah + 2a^2 \]

7. **Rearranging the Equation**:
- Rearranging gives:
\[ 4ah - 2h + 2a^2 - 5 = 0 \]
- Factor out \( h \):
\[ h(4a - 2) + (2a^2 - 5) = 0 \]

8. **Finding the Locus**:
- For the locus to be valid for all \( h \), the coefficient of \( h \) must be zero:
\[ 4a - 2 = 0 \implies a = \frac{1}{2} \]
- Substitute \( a = \frac{1}{2} \) into \( 2a^2 - 5 = 0 \):
\[ 2\left(\frac{1}{2}\right)^2 - 5 = 0 \implies \frac{1}{2} - 5 = 0 \implies -\frac{9}{2} \neq 0 \]
- Thus, we need to find when \( 2a^2 - 5 = 0 \):
\[ 2a^2 = 5 \implies a^2 = \frac{5}{2} \implies a = \pm \sqrt{\frac{5}{2}} \]

9. **Sum of All Possible Values of \( a \)**:
- The possible values of \( a \) are \( \sqrt{\frac{5}{2}} \) and \( -\sqrt{\frac{5}{2}} \).
- Therefore, the sum of all possible values of \( a \) is:
\[ \sqrt{\frac{5}{2}} + (-\sqrt{\frac{5}{2}}) = 0 \]

### Final Answer:
The sum of all possible values of \( a \) is \( 0 \).