If `(x_(1),y_(1))` and `(x_(2),y_(2))` are ends of focal chord of `y^(2)=4ax` then `x_(1)x_(2)+y_(1)y_(2)` =

Options

A. `a^(2)`
B. `-3 a^(2)`
C. `5a^(2)`
D. `-5a^(2)`

Answer

To solve the problem, we need to find the value of \( x_1 x_2 + y_1 y_2 \) for the ends of a focal chord of the parabola given by the equation \( y^2 = 4ax \).

### Step-by-Step Solution:

1. **Identify the coordinates of the points on the parabola**:
The coordinates of the points on the parabola can be expressed in terms of the parameter \( t \). For a focal chord, we can denote the coordinates of the points as:
\[ P(t_1) = (at_1^2, 2at_1) \quad \text{and} \quad Q(t_2) = (at_2^2, 2at_2) \]

2. **Use the property of focal chords**:
For points \( P(t_1) \) and \( Q(t_2) \) to be ends of a focal chord, the relationship between \( t_1 \) and \( t_2 \) is given by:
\[ t_1 t_2 = -1 \]

3. **Calculate \( x_1 x_2 \)**:
The \( x \)-coordinates of the points are \( x_1 = at_1^2 \) and \( x_2 = at_2^2 \). Therefore:
\[ x_1 x_2 = (at_1^2)(at_2^2) = a^2 t_1^2 t_2^2 \]
Since \( t_1 t_2 = -1 \), we have:
\[ t_1^2 t_2^2 = (t_1 t_2)^2 = (-1)^2 = 1 \]
Thus:
\[ x_1 x_2 = a^2 \cdot 1 = a^2 \]

4. **Calculate \( y_1 y_2 \)**:
The \( y \)-coordinates of the points are \( y_1 = 2at_1 \) and \( y_2 = 2at_2 \). Therefore:
\[ y_1 y_2 = (2at_1)(2at_2) = 4a^2 t_1 t_2 \]
Again, using \( t_1 t_2 = -1 \):
\[ y_1 y_2 = 4a^2 \cdot (-1) = -4a^2 \]

5. **Combine \( x_1 x_2 \) and \( y_1 y_2 \)**:
Now, we can find \( x_1 x_2 + y_1 y_2 \):
\[ x_1 x_2 + y_1 y_2 = a^2 - 4a^2 = -3a^2 \]

### Final Result:
Thus, the value of \( x_1 x_2 + y_1 y_2 \) is:
\[ \boxed{-3a^2} \]