A parabola is a graph of a quadratic function. Pascal stated that a parabola is a projection of a circle. Galileo explained that projectiles falling under the effect of uniform gravity follow a path called a parabolic path. Many physical motions of bodies follow a curvilinear path which is in the shape of a parabola.
In mathematics, any plane curve that is mirror-symmetrical and usually of approximately U shape is called a parabola. Here we shall aim at understanding the derivation of the standard formula of a parabola, the different equations of a parabola, and the properties of a parabola.
What is Parabola?
A parabola refers to an equation of a curve, such that each point on the curve is equidistant from a fixed point and a fixed line. The fixed point is called the “focus” of the parabola, and the fixed line is called the “directrix” of the parabola. Also, an important point to note is that the fixed point does not lie on the fixed line. Thus, a parabola is mathematically defined as follows:
“A locus of any point which is equidistant from a given point (focus) and a given line (directrix) is called a parabola.”
A parabola is an important curve of the conic sections of the coordinate geometry.
Parabola Examples
Example 1: The equation of a parabola is y2 = 24x. Find the length of the latus rectum, focus, and vertex.
Solution:
To find: Length of latus rectum, focus and vertex of the parabola
Given: Equation of a parabola: y2 = 24x
Therefore, 4a = 24
a = 24/4 = 6
Now, parabola formula for latus rectum is:
Length of latus rectum = 4a
= 4(6) = 24
Now, focus= (a,0) = (6,0)
Now, Vertex = (0,0)
Answer: Length of latus rectum = 24, focus = (6,0), vertex = (0,0)
Example 2: The equation of a parabola is 2(y-3)2 + 24 = x. Find the length of the latus rectum, focus, and vertex.
Solution:
To find: the length of the latus rectum, focus, and vertex of a parabola
Given: equation of a parabola: 2(y-3)2 + 24 = x
On comparing it with the general parabola equation, x = a(y-k)2 + h, we get
a = 2
Length of latus rectum = 1/a = 1/2
Vertex, (h, k) = (24, 3)
Focus = (h + 1/4a, k) = (24 + 1/8, 3) = (193/8, 3)
Answer: Length of latus rectum = 1/2, focus = (193/8, 3), Vertex = (24,3)
Example 3. Which equation represents a parabola that has a focus of (0, 0) and a directrix of y = 4?
Solution:
Given that, Focus = (0, 0) and directrix y = 4
Let us suppose that there is a point (x, y) on the parabola.
Its distance from the focus point (0, 0) is √((x − 0)2 + (y – 0)2 )
Its distance from directrix y = 4 is |y – 4|
By the definition of a parabola, these two distances are the same.
√[(x − 0)2 + (y – 0)2] = |y – 4|
Squaring on both sides.
(x − 0)2 + (y – 0)2 = (y – 4)2
x2 + y2 = y2 – 8y + 16
x2 + 8y – 16 = 0
Answer: Hence, the equation of a parabola with a focus at (0, 0) and a directrix of y = 4 is x2 + 8y – 16 = 0.
Parabola Examples in Real Life
- Kicking the ball: When you kick a soccer ball, it arcs up into the air and comes down again, following the path of a parabola.
- Shooting an arrow: When you shoot an arrow, it arcs up into the air and comes down again, following the path of a parabola.
- Throwing a stone: When you throw a stone, it arcs up into the air and comes down again, following the path of a parabola.
- Satellite Dishes: A satellite dish is a type of parabolic antenna that receives or transmits information by radio waves to or from a communication satellite.
- Parabolic trough: Parabolic troughs are solar thermal collectors that are curved in three dimensions as a parabola, lined with polished metal mirrors. These are many used to concentrate the sun’s rays to make a hot spot.
- Spotlight reflectors: Spotlights have parabolic reflectors that are used to project a bright beam of light onto a performance space.
- Flashlight: A flashlight or torch has a parabolic reflector that gives a variable-focus effect from a wide floodlight to a narrow beam.
- Parabolic microphone: A parabolic microphone has a parabolic reflector that collects and focuses sound waves onto a transducer.
- Automobile headlight: Automobile headlamps have parabolic reflectors that collect and focus the beam of light to illuminate the road ahead.
- Ballistic missile: A Ballistic missile delivers warheads on a target by using projectile motion. The trajectory of these missiles makes a parabolic path.
- Jump of a Dolphin: The jump of a dolphin is known as porpoising. According to some research, dolphins use their jump as a method of communication. If you look at a dolphin’s porpoising, you will observe that they trace a parabolic path while performing the jump.
- Camera: Many cameras use an assembly of parabolic mirrors to take wide-angle shots.
- IR Spectrometer: Some gated spectrometers use a pair of 90-degree off-axis parabolic mirrors to relay the light from an entrance slit to an output IR recording camera.
- Parabolic dunes: Parabolic dunes are formed when strong winds erode a section of the vegetated sand (commonly referred to as a blowout). Parabolic dunes are common in the sand sheet southwest of the main dune field.
- Reflecting telescopes: Some reflecting telescopes use parabolic mirrors to reduce spherical aberration with no chromatic aberration.
- Parabolic Wifi antenna: A parabolic wifi device uses a parabolic antenna that is backed with a parabolic reflector that directs waves, in this case, wifi waves, to the antenna, enhancing the wifi signal.