If the ends of a focal chord of the parabola `y^(2) = 8x` are `(x_(1), y_(1))` and `(x_(2), y_(2))`, then `x_(1)x_(2) + y_(1)y_(2)` is equal to

Options

A. 12
B. 20
C. 0
D. -12

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 \( y^2 = 8x \).

### Step 1: Identify the parameters of the parabola
The given parabola is \( y^2 = 8x \). This can be rewritten in standard form as \( y^2 = 4ax \) where \( 4a = 8 \). Thus, we find \( a = 2 \).

### Step 2: Determine the coordinates of the points on the parabola
Using the parametric form of the parabola, the coordinates of points \( A \) and \( B \) can be expressed as:
- For point \( A \) (with parameter \( t_1 \)):
\[ (x_1, y_1) = (at_1^2, 2at_1) = (2t_1^2, 4t_1) \]
- For point \( B \) (with parameter \( t_2 \)):
\[ (x_2, y_2) = (at_2^2, 2at_2) = (2t_2^2, 4t_2) \]

### Step 3: Use the property of focal chords
For a focal chord, the relationship between the parameters is given by:
\[ t_1 t_2 = -1 \]
From this, we can express \( t_2 \) in terms of \( t_1 \):
\[ t_2 = -\frac{1}{t_1} \]

### Step 4: Substitute \( t_2 \) into the coordinates of point \( B \)
Substituting \( t_2 \) into the coordinates of point \( B \):
- For \( x_2 \):
\[ x_2 = 2t_2^2 = 2\left(-\frac{1}{t_1}\right)^2 = 2\frac{1}{t_1^2} = \frac{2}{t_1^2} \]
- For \( y_2 \):
\[ y_2 = 4t_2 = 4\left(-\frac{1}{t_1}\right) = -\frac{4}{t_1} \]

### Step 5: Calculate \( x_1 x_2 + y_1 y_2 \)
Now we can calculate \( x_1 x_2 + y_1 y_2 \):
\[ x_1 x_2 = (2t_1^2) \left(\frac{2}{t_1^2}\right) = 4 \]
\[ y_1 y_2 = (4t_1) \left(-\frac{4}{t_1}\right) = -16 \]
Thus,
\[ x_1 x_2 + y_1 y_2 = 4 - 16 = -12 \]

### Final Answer
The value of \( x_1 x_2 + y_1 y_2 \) is \( -12 \).

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