Education

Introduction To A Parabola And To Find The Vertex, Focus And Directrix Of The Parabola

Find The Vertex, Focus And Directrix Of The Parabola (1)

A plane curve where any point is equidistant from a fixed point called the focus and the fixed straight line called the directrix is defined as a parabola. It is a U-shaped curve.

A parabola is also defined as a conic section that is obtained from the intersection of a right circular surface of a cone and a plane that is parallel to the other plane which is being tangential to the surface of a cone.

The Parts Of A Parabola Include:

Find The Vertex, Focus And Directrix Of The Parabola

  • Line of symmetry: A line that passes through the focus and perpendicular to the directrix.
  • Vertex: It is a point at which the parabola intersects the line of symmetry. The parabola is sharply curved at this point.
  • Focal length: It is the distance measured along the axis of symmetry between the vertex and the focus.
  • Latus rectum: The chord of the parabola that passes through the focus and parallel to the directrix.

The general equation of a parabola is y2 = 4ax, where a is the distance between the origin and the focus. The vertex form of the parabola is y = a (x – h)2 + k.

 

To Find The Vertex, Focus And Directrix Of The Parabola

The standard equation of the parabola is of the form ax2 + bx + c = 0. If a > 0 in ax2 + bx + c = 0, then the parabola is opening upwards and if a < 0, then the parabola is opening downwards.

 

Vertex Of The Parabola

It is a point (h, k) on the parabola. It becomes either the base or the top depending upon the orientation of the parabola [opening upward or downward].
The point h = (–b / 2a)
The point k = [4ac – b2] / 4a

 

Focus Of The Parabola

The axis of symmetry of the parabola is a vertical line that cuts the parabola into half and is given by x = h.
It lies on the axis of symmetry of the parabola at F (h, k + p) where p = 1 / 4a.

 

You May Also Like – What Is The PH Of A 3.4×10-8 M HClO4 Solution At Room Temperature?

 

Directrix Of The Parabola

The line which is opposite to the focus and on the side of the vertex with an equation y = k – p is the directrix of the parabola.

Example 1: How to find the vertex, focus and directrix of the parabola y = x² – 6x + 15.

Solution:

The x-coordinate of the vertex is x = h = (–b / 2a)
The standard equation of the parabola is of the form ax2 + bx + c = 0.
On comparing, a = 1, b = -6, and c = 15.
x = -b / (2a)
=’yoast-text-mark’>= – (-6) / (2 * 1)
=’yoast-text-mark’>= 6 / 2
=’yoast-text-mark’>= 3
The x-coordinate of the vertex is y = k = [4ac – b2] / 4a
= [4 * 1 * 15 – (-6)2] / [4 * 1] = [24 / 4] = 6
The vertex is (3, 6).
The focus is F (h, k + p) where p = 1 / 4a.
The vertex is (3, 6) and hence the axis of symmetry is x =’yoast-text-mark’>= 3.
The x-coordinate of the focus is 3.
The y-coordinate of the focus is k + [1 / (4a)] = 6 + [1 / (4 * 1)] = 6 + [1 / 4] = 6 + 0.25
= 6.25
The focus is (3, 6.25).
The equation y = k – p is the directrix of the parabola.
y = k – p
= 6 – (1 / 4)
= 5.75

 

Applications Of A Parabola

Approximations of parabolas can be found in the shape of the main cables on a simple suspension bridge.

Paraboloids can be seen in the surface of a liquid confined to a container and when rotated around the central axis.

The vertical curves in roads are usually parabolic by design in the United States.

About the author

Editor N4GM

He is the Chief Editor of n4gm. His passion is SEO, Online Marketing, and blogging. Sachin Sharma has been the lead Tech, Entertainment, and general news writer at N4GM since 2019. His passion for helping people in all aspects of online technicality flows the expert industry coverage he provides. In addition to writing for Technical issues, Sachin also provides content on Entertainment, Celebs, Healthcare and Travel etc... in n4gm.com.

Leave a Comment