Question

The latus rectum of a parabola is a line (a) through the focus (b) parallel to the directrix (c) perpendicular to the axis (d) all of these

Short answer

The latus rectum of a parabola is a line segment that passes through the focus of the parabola, is parallel to the directrix, and is perpendicular to the axis of the parabola. Therefore, option (d) "all of these" is the correct answer, as it includes options (a), (b), and (c).

Step-by-step solution

Verify option (a)

Recall the definition of the latus rectum: it is a line segment that passes through the focus of the parabola. Therefore, option (a) is true.

Verify option (b)

The latus rectum is also parallel to the directrix. This can be visualized if we recall that both the latus rectum and the directrix are horizontal (or vertical) lines when considering the standard form of a parabola. Therefore, option (b) is true.

Verify option (c)

As the latus rectum passes through the focus and is parallel to the directrix, it is perpendicular to the axis of the parabola, which is a vertical line passing through the vertex and the focus. Therefore, option (c) is true.

Verify option (d)

Since options (a), (b), and (c) are all true, option (d) is also correct. The latus rectum of a parabola has all the properties mentioned in the exercise: it passes through the focus, is parallel to the directrix, and is perpendicular to the axis of the parabola.

Key concepts

Understanding the Focus of a Parabola

The focus is a crucial point in defining the shape and orientation of a parabola. A parabola is a curve that opens up in a way so that every point on it is equidistant from a fixed point (the focus) and a fixed line (the directrix). The focus lies inside the curve of the parabola.

It acts as a part of the parabola's geometric attributes:

• A point of reference from which distances are measured to form the curve.
• Positioned at (h, k + p) for a vertical parabola, where \(p\) is the distance from the vertex and \(k\) is its y-coordinate.
• It ensures that the parabola is symmetric about its axis.

In short, the placement of the focus helps determine how 'wide' or 'narrow' a parabola will appear. Any change in the position of the focus, relative to the vertex and directrix, will alter the parabola's opening.

Explaining the Directrix of a Parabola

The directrix is another essential component that helps in forming a parabola. It is a straight line that sits outside and opposite to the focus.

Here are some key points about the directrix:

• It is a line such that the distance from any point on the parabola to the focus is equal to the perpendicular distance from that point to the directrix.
• For a vertical parabola, this is a horizontal line described by y = k - p, opposite where the focus is located.
• Acts as the diametric counterpart to the focus in the parabola's definition.

This relationship between the value of points on the parabola to both the focus and the directrix ensures a consistent and smooth curve.

Defining the Axis of a Parabola

The axis of a parabola is a vital imaginary line that runs through the parabola's vertex. It serves as a line of symmetry for the curve.

Important features of the axis include:

• For vertical parabolas, it is a vertical line described by the equation x = h, where \(h\) is the x-coordinate of the vertex.
• The axis divides the parabola into two mirror-image halves.
• It provides a vertical reference from which both the directrix and the focus can be measured, ensuring that they are perfectly aligned.

Understanding the axis of a parabola is essential, as it not only aids in graphing the parabola but also helps in solving related algebraic problems effectively.