If the tangents at `t_(1)` and `t_(2)` on `y^(2)` = 4ax makes complimentary angles with axis then `t_(1)t_(2)`=

Options

A. 0
B. 1
C. `-1`
D. `-2`

Answer

To solve the problem, we need to find the product \( t_1 t_2 \) given that the tangents at points \( t_1 \) and \( t_2 \) on the parabola \( y^2 = 4ax \) make complementary angles with the x-axis.

### Step-by-Step Solution:

1. **Identify the Parabola Equation:**
The given equation of the parabola is:
\[ y^2 = 4ax \]

2. **Find the Points on the Parabola:**
The points corresponding to parameters \( t_1 \) and \( t_2 \) on the parabola are:
\[ P_1 = (at_1^2, 2at_1) \quad \text{and} \quad P_2 = (at_2^2, 2at_2) \]

3. **Write the Equation of the Tangents:**
The equation of the tangent at point \( P_1 \) is:
\[ y = \frac{x}{t_1} + at_1 \]
Similarly, the equation of the tangent at point \( P_2 \) is:
\[ y = \frac{x}{t_2} + at_2 \]

4. **Determine the Slopes of the Tangents:**
The slopes of the tangents at points \( P_1 \) and \( P_2 \) are:
\[ m_1 = \frac{1}{t_1} \quad \text{and} \quad m_2 = \frac{1}{t_2} \]

5. **Use the Condition of Complementary Angles:**
Since the tangents make complementary angles with the x-axis, we have:
\[ \alpha + \beta = 90^\circ \]
This implies:
\[ \tan(\alpha + \beta) = \tan(90^\circ) \quad \text{which is undefined} \]

6. **Apply the Tangent Addition Formula:**
Using the tangent addition formula:
\[ \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \]
Setting this equal to infinity means the denominator must be zero:
\[ 1 - \tan \alpha \tan \beta = 0 \]
Thus:
\[ \tan \alpha \tan \beta = 1 \]

7. **Substituting the Slopes:**
Substituting the values of the slopes:
\[ m_1 \cdot m_2 = \frac{1}{t_1} \cdot \frac{1}{t_2} = 1 \]
This simplifies to:
\[ \frac{1}{t_1 t_2} = 1 \]

8. **Solve for \( t_1 t_2 \):**
Therefore, we have:
\[ t_1 t_2 = 1 \]

### Final Answer:
\[ t_1 t_2 = 1 \]