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If the chord joining t(1) and t(2) on th...

If the chord joining `t_(1)` and `t_(2)` on the parabola `y^(2)` = 4ax is a focal chord then

A

`t_(1)t_(2)=-1`

B

`t_(1)t_(2)=1`

C

`t_(1)t_(2)=2`

D

`t_(1)t_(2)=-2`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to determine the condition for the chord joining points \( t_1 \) and \( t_2 \) on the parabola \( y^2 = 4ax \) to be a focal chord. ### Step-by-Step Solution: 1. **Understand the Parabola**: The given parabola is \( y^2 = 4ax \). The focus of this parabola is at the point \( (a, 0) \). 2. **Parametric Form**: The points on the parabola can be expressed in parametric form as: \[ P(t_1) = (at_1^2, 2at_1) \quad \text{and} \quad Q(t_2) = (at_2^2, 2at_2) \] 3. **Focal Chord Condition**: A chord is a focal chord if the product of the parameters \( t_1 \) and \( t_2 \) is equal to -1: \[ t_1 t_2 = -1 \] 4. **Finding the Slope**: The slope of the line segment joining points \( P(t_1) \) and \( Q(t_2) \) is given by: \[ \text{slope} = \frac{2at_2 - 2at_1}{at_2^2 - at_1^2} = \frac{2a(t_2 - t_1)}{a(t_2^2 - t_1^2)} = \frac{2(t_2 - t_1)}{t_2^2 - t_1^2} \] 5. **Using the Difference of Squares**: The difference of squares can be factored: \[ t_2^2 - t_1^2 = (t_2 - t_1)(t_2 + t_1) \] Thus, the slope simplifies to: \[ \text{slope} = \frac{2}{t_2 + t_1} \] 6. **Condition for Focal Chord**: For the chord to be a focal chord, the slopes from both ends of the chord must satisfy the condition derived from the properties of the parabola: \[ t_1 t_2 = -1 \] 7. **Conclusion**: Therefore, the condition for the chord joining points \( t_1 \) and \( t_2 \) on the parabola \( y^2 = 4ax \) to be a focal chord is: \[ t_1 t_2 = -1 \] ### Final Answer: The correct option is \( t_1 t_2 = -1 \). ---
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