The angle subtended by the double ordinate of length 8a of the parabola `y^(2)=4ax` at its vertex is a) `(pi)/(3)` b) `(pi)/(4)` c) `(pi)/(2)` d) `(pi)/(6)`

Options

A. `(pi)/(3)`
B. `(pi)/(4)`
C. `(pi)/(2)`
D. `(pi)/(6)`

Answer

To solve the problem, we need to find the angle subtended by the double ordinate of length \(8a\) of the parabola \(y^2 = 4ax\) at its vertex.

### Step-by-Step Solution:

1. **Understanding the Parabola**:
The given parabola is \(y^2 = 4ax\). The vertex of this parabola is at the origin \((0, 0)\).

**Hint**: Remember that the vertex of a parabola in the form \(y^2 = 4ax\) is always at the point \((0, 0)\).

2. **Identifying the Double Ordinate**:
A double ordinate is a line segment parallel to the directrix of the parabola that intersects the parabola at two points. The length of the double ordinate given is \(8a\).

**Hint**: A double ordinate can be visualized as a horizontal line segment across the parabola.

3. **Finding the Points on the Parabola**:
Let the points where the double ordinate intersects the parabola be \(P\) and \(Q\). Since the length of the double ordinate is \(8a\), we can set the coordinates of these points as:
- \(P(4a, h)\)
- \(Q(4a, -h)\)

The distance between these two points is \(h - (-h) = 2h\). We know that this distance equals \(8a\), so:
\[ 2h = 8a \implies h = 4a \]

**Hint**: Use the properties of the parabola to find the coordinates of the points where the double ordinate intersects.

4. **Calculating the Angle at the Vertex**:
The angle subtended at the vertex \(O\) by the points \(P\) and \(Q\) can be calculated using the tangent of the angle. The coordinates of \(P\) and \(Q\) are:
- \(P(4a, 4a)\)
- \(Q(4a, -4a)\)

The angle \(\theta\) can be found using the tangent function:
\[ \tan(\theta) = \frac{y}{x} = \frac{4a}{4a} = 1 \]
Therefore, \(\theta = 45^\circ\) or \(\frac{\pi}{4}\) radians.

**Hint**: Use the tangent function to relate the angle to the coordinates of the points.

5. **Finding the Total Angle Subtended**:
The total angle subtended by the double ordinate \(PQ\) at the vertex \(O\) is \(2\theta\):
\[ \text{Total angle} = 2 \times 45^\circ = 90^\circ = \frac{\pi}{2} \]

**Hint**: Remember that the angle subtended by a line segment at a point is twice the angle subtended by each endpoint at that point.

6. **Conclusion**:
The angle subtended by the double ordinate of length \(8a\) at the vertex of the parabola \(y^2 = 4ax\) is \(\frac{\pi}{2}\).

Therefore, the answer is **option (c) \(\frac{\pi}{2}\)**.