All chords of the parabola `y^(2)=4x` which subtend right angle at the origin are concurrent at the point.

Options

A. (1, 0)
B. (2, 0)
C. (3, 0)
D. (4, 0)

Answer

To solve the problem of finding the point at which all chords of the parabola \( y^2 = 4x \) that subtend a right angle at the origin are concurrent, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Parabola**:
The given parabola is \( y^2 = 4x \). This is a standard form of a parabola that opens to the right.

2. **Parameterize Points on the Parabola**:
Let \( P(t_1) \) and \( Q(t_2) \) be two points on the parabola. The coordinates of these points can be expressed as:
\[ P(t_1) = (at_1^2, 2at_1) \quad \text{and} \quad Q(t_2) = (at_2^2, 2at_2) \]
Here, \( a = 1 \) for our parabola \( y^2 = 4x \).

3. **Find the Slopes of OP and OQ**:
The slopes of the lines \( OP \) and \( OQ \) from the origin to points \( P \) and \( Q \) are:
\[ \text{slope of } OP = \frac{2at_1}{at_1^2} = \frac{2}{t_1} \]
\[ \text{slope of } OQ = \frac{2at_2}{at_2^2} = \frac{2}{t_2} \]

4. **Condition for Right Angle**:
For the lines \( OP \) and \( OQ \) to subtend a right angle at the origin, the product of their slopes must equal -1:
\[ \frac{2}{t_1} \cdot \frac{2}{t_2} = -1 \]
Simplifying gives:
\[ \frac{4}{t_1 t_2} = -1 \quad \Rightarrow \quad t_1 t_2 = -4 \]

5. **Equation of Chord PQ**:
The equation of the chord \( PQ \) can be derived using the two-point form:
\[ y - 2at_1 = \frac{(2at_2 - 2at_1)}{(at_2^2 - at_1^2)}(x - at_1^2) \]
Simplifying this gives:
\[ y - 2t_1 = \frac{(t_2 - t_1)(t_2 + t_1)}{(t_2 - t_1)(t_2 + t_1)}(x - t_1^2) \]

6. **Finding the Concurrent Point**:
After simplification, we find that the chords will meet at a fixed point. By substituting \( t_1 t_2 = -4 \) into the equation, we can derive that the point of concurrency is:
\[ (4, 0) \]

### Conclusion:
Thus, all chords of the parabola \( y^2 = 4x \) that subtend a right angle at the origin are concurrent at the point \( (4, 0) \).