The tangents drawn at the extremities of a focal chord of a parabola meet on :

Question

Accepted Answer

The correct answer to the question is Option 2: the directrix. Let's explore why this is the case and why the other options are incorrect.

### Explanation for Option 2:

In the context of a parabola, a focal chord is a line segment that passes through the focus of the parabola and has its endpoints on the parabola itself. A fundamental property of parabolas is that the tangents drawn at the endpoints of a focal chord intersect on the directrix of the parabola.

**Why does this happen?**

1. **Geometric Properties of Parabolas**: A parabola is defined as the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). This unique property leads to several interesting geometric results, one of which is the behavior of tangents at the endpoints of a focal chord.

2. **Reflective Property**: The reflective property of parabolas states that any ray coming parallel to the axis of symmetry and reflecting off the parabola will pass through the focus. Conversely, rays emanating from the focus will reflect off the parabola and travel parallel to the axis. This symmetry and reflective property ensure that the tangents at the endpoints of a focal chord will meet on the directrix.

### Explanation for Why Other Options are Incorrect:

1. **Option 1: The axis of the parabola**:
   - The axis of the parabola is the line that passes through the vertex and the focus, serving as the line of symmetry. While the axis is crucial for defining the parabola's orientation, the tangents at the endpoints of a focal chord do not necessarily intersect on this line. Instead, they intersect on the directrix, which is perpendicular to the axis.

2. **Option 3: The tangent at the vertex**:
   - The tangent at the vertex of a parabola is a horizontal line (for a standard parabola opening upwards or downwards) and does not intersect with the tangents at the endpoints of a focal chord. The tangents at the extremities of a focal chord are inclined and meet on the directrix, not on the tangent at the vertex.

3. **Option 4: Nowhere**:
   - This option suggests that the tangents do not meet at any specific location, which contradicts the established geometric property of parabolas. The tangents at the endpoints of a focal chord do indeed meet at a specific location, which is the directrix.

In conclusion, the correct answer is Option 2: the directrix, due to the inherent geometric properties of parabolas that dictate the behavior of tangents at the endpoints of a focal chord.