If the tangents at `t_(1)`and`t_(2)` on `y^(2)` = 4ax are at right angles if

Options

A. `t_(1)t_(2)=-1`
B. `t_(1)t_(2)=1`
C. `t_(1)t_(2)=2`
D. `t_(1)t_(2)=-2`

Answer

To find the condition under which the tangents at \( t_1 \) and \( t_2 \) on the parabola \( y^2 = 4ax \) are at right angles, we can follow these steps:

### Step 1: Understand the Tangent Equation
The equation of the tangent to the parabola \( y^2 = 4ax \) at a point corresponding to the parameter \( t \) is given by:
\[ y = 2a t + \frac{a}{t}(x - at^2) \]

### Step 2: Identify the Tangents
For the parameters \( t_1 \) and \( t_2 \), the equations of the tangents can be written as:
1. \( y = 2a t_1 + \frac{a}{t_1}(x - at_1^2) \)
2. \( y = 2a t_2 + \frac{a}{t_2}(x - at_2^2) \)

### Step 3: Find the Slopes of the Tangents
The slope of the tangent at \( t_1 \) is:
\[ m_1 = \frac{dy}{dx} = \frac{2a}{a/t_1} = 2t_1 \]
The slope of the tangent at \( t_2 \) is:
\[ m_2 = \frac{dy}{dx} = \frac{2a}{a/t_2} = 2t_2 \]

### Step 4: Condition for Right Angles
For the tangents to be at right angles, the product of their slopes must equal \(-1\):
\[ m_1 \cdot m_2 = -1 \]
Substituting the slopes:
\[ (2t_1)(2t_2) = -1 \]
This simplifies to:
\[ 4t_1 t_2 = -1 \]
Thus, we have:
\[ t_1 t_2 = -\frac{1}{4} \]

### Step 5: Analyze the Options
Now, we need to check the given options:
1. \( t_1 t_2 = -1 \)
2. \( t_1 t_2 = 1 \)
3. \( t_1 t_2 = 2 \)
4. \( t_1 t_2 = -2 \)

None of these options match \( t_1 t_2 = -\frac{1}{4} \). However, the condition for tangents at right angles is that the product of the parameters should be equal to \(-1\) when multiplied by \(4\).

### Conclusion
The correct condition for the tangents at \( t_1 \) and \( t_2 \) on the parabola \( y^2 = 4ax \) to be at right angles is:
\[ t_1 t_2 = -1 \]