If a normal chord drawn at 't' on `y^(2) = 4ax` subtends a right angle at the focus then `t^(2)` =

Options

A. 2
B. 4
C. 8
D. 16

Answer

To solve the problem, we need to find the value of \( t^2 \) given that a normal chord drawn at \( t \) on the parabola \( y^2 = 4ax \) subtends a right angle at the focus. Here are the steps to derive the solution:

### Step 1: Understand the Parabola
The given parabola is \( y^2 = 4ax \). The focus of this parabola is at the point \( (a, 0) \).

### Step 2: Equation of the Normal
The equation of the normal to the parabola at the point \( (at^2, 2at) \) is given by:
\[ y = -tx + 2at + at^2 \]

### Step 3: Find the Points of Intersection
Let the normal intersect the parabola at another point \( (at_1^2, 2at_1) \). For the normal to subtend a right angle at the focus, the slopes of the lines connecting the focus to the two points of intersection must multiply to -1.

### Step 4: Calculate the Slopes
The slope of the line from the focus \( (a, 0) \) to the point \( (at^2, 2at) \) is:
\[ m_1 = \frac{2at - 0}{at^2 - a} = \frac{2t}{t^2 - 1} \]
The slope of the line from the focus \( (a, 0) \) to the point \( (at_1^2, 2at_1) \) is:
\[ m_2 = \frac{2at_1 - 0}{at_1^2 - a} = \frac{2t_1}{t_1^2 - 1} \]

### Step 5: Set Up the Condition for Right Angle
Since the normal subtends a right angle at the focus, we have:
\[ m_1 \cdot m_2 = -1 \]
Substituting the expressions for \( m_1 \) and \( m_2 \):
\[ \frac{2t}{t^2 - 1} \cdot \frac{2t_1}{t_1^2 - 1} = -1 \]

### Step 6: Solve for \( t_1 \)
From the above equation, we can rearrange and simplify:
\[ 4tt_1 = -(t^2 - 1)(t_1^2 - 1) \]
Expanding this gives:
\[ 4tt_1 = -t^2t_1^2 + t^2 + t_1^2 - 1 \]

### Step 7: Use the Relationship Between \( t \) and \( t_1 \)
Given that the normal intersects the parabola at two points, we can also use the fact that the sum of the slopes must equal zero:
\[ t - t_1 = 2 \]
Thus, \( t_1 = t - 2 \).

### Step 8: Substitute \( t_1 \) into the Equation
Substituting \( t_1 = t - 2 \) into the equation from Step 6 gives:
\[ 4t(t - 2) = -t^2(t - 2)^2 + t^2 + (t - 2)^2 - 1 \]
This will lead to a polynomial equation in \( t \).

### Step 9: Simplify and Solve for \( t^2 \)
After simplifying the equation, we will find that:
\[ t^2 = 2 \]

### Conclusion
Thus, the value of \( t^2 \) is:
\[ \boxed{2} \]