What comes to mind when you hear the word PARABOLA? Isn’t it true that the first thing that springs to mind is MATHEMATICS? But one thing we don’t know is the significance of the parabola. We all understand that a parabola is a basic curve. A parabolic shape plays an important role in the real world. An electric fire, a flashlight, a solar cooker, a satellite dish, and a parabolic microphone use the parabolic shape to act as transmitter or receiver. So there is a reason to use parabolic shape. Let us dive in to learn about it in detail.

What is Parabola?

A parabola is a section of a right circular cone created by cutting it with a plane parallel to the cone’s slant or generator. It is the locus of a moving point in a plane whose distance from a fixed point equals its distance from a fixed line that doesn’t contain the fixed point. The fixed point is called the focus, and the fixed line is called the directrix.

The axis of symmetry is the line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola in half).
The vertex is the point on the parabola where its axis of symmetry intersects, and it is also the place where the parabola is most steeply curved.
The focal length is the distance between the vertex and the focus as measured along the axis of symmetry.
The chord of the parabola that is parallel to the directrix and passes through the focus is known as the latus rectum.
Parabolas can open in any direction: up, down, left, right, or any other.

Parabola Diagram

The graph of a standard parabola \( {y^2} = 4ax\) whose vertex is \(\left({0,0} \right)\), the focus is \(\left({a,0} \right)\) and directrix is \(x=-a\) (where \(a\) is the distance between vertex and focus) are given below:

Here, \(a > 0\), the parabola is opened up on the right side. If \(a < 0\), the parabola will be opened up on the left side. All four types of standard parabola diagram as follows:

The graph of quadratic expressions is also parabolas. The graph of the quadratic expression \(y = a{x^2} + bx + c\) is a parabola that is either opening upward or downwards depending upon \(a > 0\) or \(a < 0\) , respectively. The axis of symmetry for this quadratic expression is given by \(x = \frac{ { – b}}{ {2a}}\) Hence, the coordinates of the vertex \(y = a{x^2} + bx + c\) is given by \(\left({\frac{{ – b}}{{2a}},0} \right)\) For example, Graph \(f\left(x \right) = {x^2} + 2x + 3\) and \(f\left( x \right) = – {x^2} + 2x + 3\) are shown below:

Properties of Parabola

Some of the important properties of a parabola are as follows:
1. Eccentricity is a conic section factor that indicates how round the conic section is. More eccentricity indicates less spherical behaviour, while less eccentricity indicates more spherical behaviour. It is symbolised by the letter \({\rm{e}}{\rm{.}}\) The ratio of the distance between the focus and a point on the plane to the vertex and that point only is the eccentricity of a parabola. Thus, any parabola has an eccentricity \(1\).
2. The parabola is symmetric with respect to its axis.
3. The axis runs perpendicular to the directrix.
4. The focus and the vertex are connected via the axis.
5. At the vertices, the tangent is parallel to the directrix.
6. The vertex is the focus’s midpoint and the place where the directrix and axis cross.
7. If \(a\) is the distance between focus and vertex, the distance between the focus and the point on the plane is \(2a\), which is equal to the distance between the focus and the directrix. The axis of symmetry is the halfway of the latus rectum, indicating that the half of the latus rectum is \(2a\) and the length of the latus rectum is \(4a\).

Examples of Parabola in Real-life

There are a lot of real-life examples where parabola plays an important role; some of them are:
1. When a liquid is rotated, gravity forces cause the liquid to form a parabola-like shape. The most common example is when you rotate an orange juice glass around its axis to stir it up. The juice level rises along the sides of the glass while lowering somewhat in the middle (the axis). The whirlpool is another example of whirling liquids.

2. Satellite dishes also use parabolas to help reflect signals that are subsequently sent to a receiver. For example, the National Radio Astronomy Observatory, located in Green Bank, West Virginia, is home to the world’s most advanced astronomical telescope, which operates at wavelengths ranging from centimetres to millimetres. Because of the reflecting qualities of parabolas, signals sent directly to the satellite will bounce off and return to the receiver after bouncing off the focus.

3. In the realm of architecture and engineering, parabolas are used. The parabola, a structure in London created in \(1962\) that consists of a copper roof with parabolic and hyperbolic lines, is an example of a parabolic shape. In San Francisco, California, the Golden Gate Bridge has parabolas on each side of its side spans.

4. The reflecting properties of parabolas are used in several heaters. The heat source lies in the centre, with parallel beams concentrating the heat.

5. When light needs to be focussed, parabolas are frequently used. For example, a parabolic reflector helps in focusing light into a beam that can be seen from a long distance. It aids in the reduction of light utilisation, and so improves the surface of the parabola.

Solved Examples – Parabola

Q.1. What will be the coordinates of focus and vertex for the parabola \( {y^2} = 12x\) ?
Ans: Since we know that for \({y^2} = 4ax\), coordinates of focus is \(\left({a,0} \right)\) and vertex is \(\left({0,0} \right)\) So, on comparing with \({y^2} = 12x\), we get \(4a = 12 \Rightarrow a = 3\) Hence, the coordinates of focus are \(\left({3,0} \right)\) ,and vertex is \(\left( {0,0} \right)\)

Q.2. What will be the parabola equation whose focus is \(\left({2,0} \right)\) and vertex be at the origin?
Ans: As we know, if the coordinates of focus are \(\left({a,0} \right)\) and vertex is \(\left({0,0} \right)\) then the equation of a parabola is \( {y^2} = 4ax\). Hence, for focus \(\left({2,0} \right)\) and vertex \(\left({0,0} \right)\), the required parabola \({y^2} = 8x\)

Q.3. If the distance between focus and vertex of a parabola is \(10\,{\text{cm}}\). What will be the length of the latus rectum of that parabola?
Ans: Since we know that, If \(a\) is the distance between focus and vertex, then the distance between the focus and the point on the plane is \(2a\), which is equal to the distance between the focus and the directrix. The axis of symmetry is the halfway of the latus rectum, indicating that the half of the latus rectum is \(2a\) and the length of the latus rectum is \(4a\). So, Here \(a = 10\,{\text{cm}} \Rightarrow \) Length of latus rectum is \(4 \times 10\,{\text{cm=40}}\,{\text{cm}}\)

Q.4. What will be the coordinates of the vertex and \(y – \) intercept for the parabola \(y = {x^2} + 2x + 3?\)
Ans: As we know, for the parabola \(y = a{x^2} + bx + c\), the coordinates of the vertex is \(\left({\frac{{ – b}}{{2a}},0} \right)\). Hence, for parabola \(y = {x^2} + 2x + 3\), the coordinates of the vertex is \(\left({\frac{{ – 2}}{{2 \times 1}},0} \right) = \left({ – 1,0} \right)\). To find \(y – \) intercept we put \(x = 0\) in the parabola \(y = {x^2} + 2x + 3\), which will give the value of \(y – \) intercept. So, \(y – \)intercept \( = {0^2} + 2 \times 0 + 3 = 3\) \( \Rightarrow \) coordinates of \(y – \) intercept is \(\left({0,3} \right)\)

Q.5. If the parabola \(y = {x^2} + bx + c\), passes through the two points \(\left({1,6} \right)\) and \(\left({0,3} \right)\) What will be the value of \(b\) and \(c\)?
Ans: Since the parabola passes through two points, the two points will satisfy the parabola equation.
For parabola \(y = {x^2} + bx + c\),
\( \Rightarrow 6 = {1^2} + b \times 1 + c\) and \(3 = {0^2} + b \times 0 + c\)
\( \Rightarrow 5 = b + c\) and \(c = 3\)
\( \Rightarrow 5 = b + 3\)
\( \Rightarrow b = 2\)

Summary

Parabola is not simply about mathematics or how we plot a parabola. Parabola has its own significance, which only a few understand, significance that can produce something extraordinary, and significance that can delight others with its beauty and uniqueness in buildings. We have seen its immense uses in the real world, which led to a significant role in the mathematical world.

Frequently Asked Questions (FAQ) – Parabola

Q.1. What is the definition of the parabola? Give an example.
Ans: A parabola is a symmetrical plane curve formed when a cone intersects a plane parallel to one of its sides. The U-shaped graph of a quadratic expression is an example of a parabola.

Q.2. Is rainbow a parabola?
Ans: A rainbow isn’t shaped like a parabola. It’s a part of a circle. The rainbow can be seen anywhere where the angle between the direct light from the sun and the refractured light reaching your eyes is fixed.

Q.3. What is a parabola shape?
Ans:  A parabola is a mirror-symmetrical plane curve that is roughly U-shaped in mathematics. It fits various multifarious mathematical descriptions, all of which can be shown to define the same curves. A parabola has a point (the focus) and a line (the directrix).

Q.4. What is the equation of the parabola whose vertex is origin and focus lies on positive \(x\)-axis?
Ans: \({y^2} = 4ax\) is the parabola equation with vertex at the origin and focus at \(\left({a,0} \right)\) where \(a > 0\).

Q.5. Where is the vertex of a parabola?
Ans: The vertex of a parabola is the point at which the parabola and its symmetry line intersect. The vertex of a parabola whose equation in standard form as \(y = a{x^2} + bx + c\) will be the graph’s minimum (lowest point) if \(a > 0\) and the graph’s maximum (highest point) if \(a < 0\).

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