Relationship Between Focus and Directrix

Oct 13, 2025 by miodeya.com 70 views

Overview of Parabolas

A parabola is a type of quadratic curve that has a specific shape, resembling a bowl or a U-shape. It is defined as the set of all points that are equidistant to the focus and the directrix. The focus is a fixed point, and the directrix is a fixed line. The parabola is symmetric about its axis, which passes through the vertex.

Properties of Parabolas

The vertex of a parabola is the point where the parabola changes direction. It is the lowest or highest point on the parabola, depending on the orientation of the parabola. The focus of a parabola is a fixed point that is equidistant to the directrix and the parabola. The directrix is a fixed line that is perpendicular to the axis of the parabola.

Parabolas with Vertex at (0,0)

A parabola with its vertex at (0,0) has a specific orientation. Since the vertex is at the origin, the parabola opens either upwards or downwards. The focus of the parabola is on the negative part of the y-axis, which means that the parabola opens downwards.

Directrix of the Parabola

The directrix of a parabola is a fixed line that is perpendicular to the axis of the parabola. Since the parabola opens downwards, the directrix will be a horizontal line that is above the vertex. The equation of the directrix can be written as y = k, where k is a positive constant.

Equation of the Parabola

The equation of a parabola with its vertex at (0,0) and focus on the negative part of the y-axis can be written as:

y = ax^2

where a is a negative constant. The value of a determines the shape of the parabola.

Relationship Between Focus and Directrix

The focus and directrix of a parabola are related by the equation:

d = 1/(4|a|)

where d is the distance between the focus and the vertex. Since the focus is on the negative part of the y-axis, the distance d is positive.

Conclusion

Based on the properties of parabolas, we can conclude that:

The directrix will cross through the positive part of the y-axis.
The parabola opens downwards.

Discussion

The parabola with its vertex at (0,0) and focus on the negative part of the y-axis has a specific orientation and shape. The directrix is a horizontal line that is above the vertex, and the parabola opens downwards. The equation of the parabola can be written as y = ax^2, where a is a negative constant.

Final Answer

The two true statements about the parabola are:

The directrix will cross through the positive part of the y-axis.
The parabola opens downwards.

Note: The other options are not true statements about the parabola.

Overview of Parabolas

Q&A

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point where the parabola changes direction. It is the lowest or highest point on the parabola, depending on the orientation of the parabola.

Q: What is the focus of a parabola?

A: The focus of a parabola is a fixed point that is equidistant to the directrix and the parabola. It is a point that lies on the axis of the parabola.

Q: What is the directrix of a parabola?

A: The directrix of a parabola is a fixed line that is perpendicular to the axis of the parabola. It is a line that lies outside the parabola and is equidistant to the focus and the parabola.

Q: What is the equation of a parabola with its vertex at (0,0) and focus on the negative part of the y-axis?

A: The equation of a parabola with its vertex at (0,0) and focus on the negative part of the y-axis can be written as:

y = ax^2

where a is a negative constant. The value of a determines the shape of the parabola.

Q: How is the focus and directrix related?

A: The focus and directrix of a parabola are related by the equation:

d = 1/(4|a|)

where d is the distance between the focus and the vertex. Since the focus is on the negative part of the y-axis, the distance d is positive.

Q: What is the relationship between the parabola and the directrix?

A: The parabola and the directrix are related in such a way that the distance between any point on the parabola and the directrix is equal to the distance between the focus and the point on the parabola.

Q: What is the significance of the vertex in a parabola?

A: The vertex of a parabola is significant because it is the point where the parabola changes direction. It is also the lowest or highest point on the parabola, depending on the orientation of the parabola.

Q: What is the significance of the focus in a parabola?

A: The focus of a parabola is significant because it is a fixed point that is equidistant to the directrix and the parabola. It is also a point that lies on the axis of the parabola.

Q: What is the significance of the directrix in a parabola?

A: The directrix of a parabola is significant because it is a fixed line that is perpendicular to the axis of the parabola. It is also a line that lies outside the parabola and is equidistant to the focus and the parabola.

Conclusion

A parabola with its vertex at (0,0) and focus on the negative part of the y-axis has a specific orientation and shape. The directrix is a horizontal line that is above the vertex, and the parabola opens downwards. The equation of the parabola can be written as y = ax^2, where a is a negative constant. The focus and directrix are related by the equation d = 1/(4|a|), and the parabola and the directrix are related in such a way that the distance between any point on the parabola and the directrix is equal to the distance between the focus and the point on the parabola.

Final Answer

The two true statements about the parabola are:

The directrix will cross through the positive part of the y-axis.
The parabola opens downwards.

Note: The other options are not true statements about the parabola.