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:
- 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.
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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.
