If`'t_1'`and`'t_2'`be the ends of a focal chord of the parabola `y^2=4ax` then`t_1t_2`is equal to

Options

A. 1
B. -1
C. 2
D. none of these

Answer

To solve the problem, we need to find the product \( t_1 t_2 \) where \( t_1 \) and \( t_2 \) are the parameters of the ends of a focal chord of the parabola given by the equation \( y^2 = 4ax \).

### Step-by-Step Solution:

1. **Understanding the Parabola**:
The equation of the parabola is \( y^2 = 4ax \). The focus of this parabola is at the point \( (a, 0) \).

2. **Coordinates of Points on the Parabola**:
For a parameter \( t_1 \), the coordinates of the point \( P \) on the parabola are:
\[ P(t_1) = (at_1^2, 2at_1) \]
Similarly, for the parameter \( t_2 \), the coordinates of the point \( Q \) are:
\[ Q(t_2) = (at_2^2, 2at_2) \]

3. **Focal Chord Condition**:
A focal chord is a line segment that passes through the focus and has its endpoints on the parabola. The condition for \( P \) and \( Q \) to be the ends of a focal chord is that the slopes of the lines \( PQ \) and \( PS \) (where \( S \) is the focus) must be equal.

4. **Finding the Slopes**:
The slope of line \( PQ \) is given by:
\[ \text{slope of } PQ = \frac{2at_2 - 2at_1}{at_2^2 - at_1^2} = \frac{2a(t_2 - t_1)}{a(t_2^2 - t_1^2)} = \frac{2(t_2 - t_1)}{t_2^2 - t_1^2} \]
The slope of line \( PS \) (from \( P \) to the focus) is:
\[ \text{slope of } PS = \frac{2at_1 - 0}{at_1^2 - a} = \frac{2at_1}{a(t_1^2 - 1)} = \frac{2t_1}{t_1^2 - 1} \]

5. **Setting the Slopes Equal**:
Setting the slopes equal gives:
\[ \frac{2(t_2 - t_1)}{t_2^2 - t_1^2} = \frac{2t_1}{t_1^2 - 1} \]

6. **Cross-Multiplying**:
Cross-multiplying yields:
\[ 2(t_2 - t_1)(t_1^2 - 1) = 2t_1(t_2^2 - t_1^2) \]
Simplifying this leads to:
\[ (t_2 - t_1)(t_1^2 - 1) = t_1(t_2^2 - t_1^2) \]

7. **Rearranging Terms**:
Rearranging terms gives:
\[ t_2 - t_1 = \frac{t_1(t_2^2 - t_1^2)}{t_1^2 - 1} \]

8. **Finding the Product**:
From the properties of focal chords in a parabola, we know that:
\[ t_1 t_2 = -1 \]

### Final Answer:
Thus, the product of the parameters \( t_1 \) and \( t_2 \) for the ends of the focal chord is:
\[ t_1 t_2 = -1 \]