A laturectum of an ellipse is a line

Options

A. passing through a focus
B. passing through the najor axis
C. perpendicular to the major axis
D. parallel to the major axis

Answer

To solve the question regarding the properties of the latus rectum of an ellipse, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding the Definition**:
The latus rectum of an ellipse is defined as a line segment that passes through one of the foci of the ellipse, is perpendicular to the major axis, and intersects the ellipse.

2. **Drawing the Ellipse**:
Start by sketching a standard ellipse. The standard form of an ellipse centered at the origin is given by the equation:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
where \(a\) is the semi-major axis and \(b\) is the semi-minor axis.

3. **Identifying the Foci**:
The foci of the ellipse are located at the points \((\pm c, 0)\), where \(c = \sqrt{a^2 - b^2}\).

4. **Drawing the Latus Rectum**:
The latus rectum can be drawn as a vertical line segment through one of the foci. For example, if we take the focus at \((c, 0)\), the latus rectum will be a vertical line segment that intersects the ellipse.

5. **Checking the Properties**:
- **Passes through the Focus**: The latus rectum indeed passes through the focus \((c, 0)\).
- **Passes through the Major Axis**: The latus rectum intersects the major axis (the x-axis in this case) at the focus.
- **Perpendicular to the Major Axis**: The latus rectum is vertical, which means it is perpendicular to the horizontal major axis.
- **Parallel to the Minor Axis**: The latus rectum is not parallel to the major axis; rather, it is perpendicular to it.

6. **Conclusion**:
Based on the properties discussed, we conclude that:
- The latus rectum passes through the focus.
- It intersects the major axis.
- It is perpendicular to the major axis.
- It is not parallel to the major axis.

### Final Answer:
The correct properties of the latus rectum of an ellipse are:
1. It passes through the focus.
2. It intersects the major axis.
3. It is perpendicular to the major axis.