Directrices

A parabola is a group of points in a plane that are away from the same distance from a given point and a given line. Directrix refers to the line that connects the focal point of a parabola. The directrix cannot contact the parabola, as it is parallel to the axis of symmetry. If the axis of symmetry is horizontal, the directrix is vertical. The directrix is a horizontal line of the type y = c, if we only consider parabolas that open upwards or downwards. Now, let’s understand some of the key fundamentals covered under this important topic.

Vertex, focus and directrices–the relationship

• The parabola’s vertex is located at the same distance from the focus as the directrix.
• A parabola with F as its focal point, V as its vertices, and D as the point of intersection of the directrix, and the symmetry’s axis is a parabola with V as its vertex.
• Its intersection point is FD.

An example: If a parabola has a vertical axis of symmetry with its vertex at (1, 4) and its focus at (1, 2), determine the directrices equation. Solution: When the parabola’s focus is F, the vertex is V, and the symmetry’s axis intersects at D, then V is the middle point of the line segment FD. The x-coordinates need to be equated and p needs to be solved as follows:

• 1 = 1 + p/2
• 2 = 1 + p
• p = 1

The x-coordinates need to be equated and p needs to be solved as follows:

• 4 = 2 + q/2
• 8 = 2 + q
• q = 6

The directrix’s equation is y = c, and it intersects at the position (1,6). In this case, c is equal to six. As a result, y = 6 is the directrices equation.

Key characteristics of directrices

• Conic sections are important to analytical geometry and can be used in various contexts. Different shapes and qualities can be found in a conic section. A parabola is a curve created by the intersection of a cone and a plane, with the plane’s slope matching that of the cone’s side. However, the two terms “directrix” and “focus” are the most important characteristics for a formal description of a parabola.
• Here’s how it’s defined in the formal sense: The definition of parabola states that every point in a plane stands at an equal distance from two other points or lines. The focus and the directrix of the parabola are the fixed point and fixed-line, respectively, provided.
• This can be either horizontal or vertical, depending on the situation. All parabolas feature this axis of symmetry. It’s important to note that the directrix never intersects the parabola’s curve; instead, it runs parallel to it. The directrices line is horizontal if the parabola has a vertical axis of symmetry.

Determining directrices and their relations in detail

• Directrix equations are always dependent on the nature of the parabola, whether it is horizontal or vertical. Even more so, it’s divided into two sections, each of which is centred at the origin or an alternate location.
• Directrices equation depends on the focus point because it is always parallel to the line flowing through it.

Vertical Parabola (up or down), vertex, (0, 0):

Equation: y= (1/4p) x²

• or,  x2=4 py
• focus:  (0,p)
• directrix:  y=-p
• Vertex (h,k): equation: y-k=1/4p (x-h)2
• or (x-h)2= 4p(y-k)
• focus: (h, k+p), directrix: y=k-p

Vertical Parabola (left or right), vertex, (0, 0):

Equation: y= (1/4p) y²

• or,  y2=4px
• focus:  (p,0)
• directrix: x=-p
• Vertex (h,k): equation: x-h=1/4p (y-k)2
• or (y-k)2=4p(x-h)
• focus: (h+p, k)
• directrix: x=h-p

p stands for the point of focus in (x, y) where the parabola curve intersects. The midpoint of the line segment connecting the focus and directrices points is another relationship between the points of focus, directrix, and vertex.

The determination of the directrices equation

• The basics of a parabola’s directrix and its relationship to other parameters like focus and vertex have been explained. Using this information, a number of issues that will assist clarify the concept of the directrix will be addressed.
• As a result, the first challenge is to determine the directrix equation when the coordinates of the focus and vertex are provided as (1, 2) and (1, 4).
• Consider the directrix-focus-vertex connection, which stipulates that the vertex’s position will be the midpoint of the line segment combining the focus and directrix’s coordinates.

A parabola’s equation is (x+2)² = −6 (y − 1)

• As a result, the parabola’s general equation will be (x − h) ²= 4p(y−k)
• When we compare both equations=

h = −2, k = 1 and 4p = −6, i.e. p = −6/4 = −3/2

• Therefore, the directrices equation for this instance will be:
• y = k − p
• y = 1 − (−3/2)
• y = 1 + 3/2
• y = 5/2

What is the Conic Section Directrix?

As per the directrices study material, a directrix in the conic section is a line that joins the point known as the focus. Conic sections can be defined as a set of points, where the distance from the focus to the line, known as the directrix, is proportional to the distance horizontally from the line.

• A parabola is formed by a conic section with r = 1.
• A hyperbola is a conic section if r > 1.
• An ellipse is a conic section of r = 1.

Conclusion

The phenomenon of directrices has been explained along with their characteristics. We have touched upon the point as to what a directrix exactly is with the type y=c and established the relationships between vertex, focus, and directrices with a few examples. Moreover, we have also discussed the point of focus (x,y), which is the intersection point of the parabola, and this has been shown through two conditions: y= (1/4p) x2, and y= (1/4p) y2. This segment also highlights that the vertex’s position will be the midpoint of the line segment integrating the focus and directrix’s coordinates, according to the directrix-focus-vertex connection. Lastly, we have touched upon conic sections, which are made up of points where the distance between the focus and the line, known as the directrix, is proportional to the horizontal distance from the line.