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Difference Between Parabola and Hyperbola

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Last updated date: 04th Mar 2024
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Parabola vs Hyperbola

When a solid figure, cone, is cut by a plane, the section which is obtained is called a conic section. Conic sections could be circles, ellipses, hyperbolas, and parabolas depending upon the angle of intersection between the axis of the cone. So, Parabola and Hyperbola are said to be two different sections of a cone. Both parabolas and hyperbolas are not closed curves like a circle or an ellipse; they are an open curve which means that the arms or branches of the curves continue to infinity. We can deal with their differences in a very simple way.In this article will study the difference between parabola and hyperbola in a very simple way. Below figure represents the parabola and hyperbola sections.


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What is Parabola?

Parabola is approximately a U-shaped  plane curve that is mirror-symmetrical. A parabola is the curve obtained when the plane cuts parallel to the side of a cone.

In a parabola, the line perpendicular to the directrix and passing through the focus is called the axis of symmetry. 


When the parabola is intersected by the point on the “axis of symmetry,” it is referred to as the “vertex.” All parabolas are shaped identically as they are cut at a specific angle. It is characterized by the eccentricity of “1.” This is the reason why they are all the same shape but can be of different sizes.


In a parabola, the two arms of the curve, become parallel to each other at infinity. 

The parabola is given by the equation y²=X

When a set of points present in a plane are equidistant from the directrix, a given straight line, and are equidistant from the focus, a given point which is fixed, it is called a parabola.Parabolas have many practical applications. They are used for designing the path of missiles, headlight reflectors of automobiles, telescopes, radar receivers, and satellite dishes.


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What is Hyperbola

Hyperbola is the curve obtained when the plane cuts almost parallel to the axis. Hyperbolas are not identical in shape as there are many angles between the axis and the plane. The points on the two arms which are closest are called the vertices; whereas the line segment which connects the arms is called the “major axis.”


The two arms or curves in a hyperbola do not become parallel. A hyperbola’s center is the midpoint of the major axis.

Hyperbola is given by the equation XY=1


At the point when the distinction of separations between a lot of focuses present in a plane to two fixed foci or focuses is a positive constant, it is known as a hyperbola.


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Difference Between Parabola and Hyperbola

Parabola

Hyperbola

When a set of points in a plane are equidistant from a given directrix or a straight line and the focus then it is called a parabola

When the difference of distances between a set of points present in a plane to two fixed points is a positive constant, it is called a hyperbola

In a parabola the two arms on extending become parallel to each other.

They do not become parallel.

All parabolas are of the same shape irrespective of the size

All hyperbolas are of different shapes.


Parabola is given by the equation y2=X; 

The a hyperbola is given by the equation XY=1


Fun Facts

  • The word “Hyperbola'' was derived from the Greek word meaning “overthrown” or “excessive”. The term hyperbola has been coined by Apollonius of Perga.

  • The word ‘parabola’ refers to the parallelism of the conic section and the tangent of the conic mantle. Also the parable has been derived from the Greek ‘parabole’.


Solved Examples

Example 1: Find the equation of  normal at the point (6, 3) to hyperbola a²/18 − y²/9 = 1

Solution:

Equation of Normal at point (x1, y1) is a² = 18, b² = 9

a2xx1 + b2yy1 = a² + b²

Equation of Normal at point (6, 3) is

18x6+9y3=18+9

3x + 3y = 27

3(x + y) = 27

x+y=273

x + y = 9

Chord of contact:

T = 0

xx1a2yy1b2=1


Example 2: Find the equation of normal to the parabola y = 4x passing through the point (15, 12).

Solution: Equation of the normal with slope m is

y = mx - 2m - m3

It passes through the point (15,12)

12 =15m - 2m - m3

m3 - 13m + 12 = 0

(m-1) ( m - 3) ( m + 4) = 0

Hence m = 1, 3 or -4

Hence the equation is

y = x - 3 ; y = 3x -33; y = 4x = 72.

FAQs on Difference Between Parabola and Hyperbola

1. What is Conic?

Answer: A conic section is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic sections are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though historically it was sometimes called a fourth type. By taking different slices through a cone you can create a circle, an ellipse, a parabola or a hyperbola.

  • A slice through a cone parallel to an edge of the cone is a parabola:

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  • A slice through a cone at a slight angle is an ellipse:

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  • A slice through a cone at a steep angle is a hyperbola:

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  • A slice straight through a cone parallel to its base is a circle

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2. Differentiate Between Circle and Ellipse.

Answer:  When a cone is sliced, a circle and an ellipse are sections of a cone. The cone has four sections that are circle, ellipse, hyperbola, and parabola. Circles and ellipses are differentiated on the basis of the angle of intersection between the plane and the axis of the cone. Both circles and ellipses are closed curves.


In a circle, all the diameters are the same size, but in an ellipse there are major and minor axes which are of different lengths.


Similarly, the radii of a circle are all the same length. In an ellipse, the semi-major axis and semi-minor axis are of different lengths.

The eccentricity of a circle is always 1; the eccentricity of an ellipse is 0 to 1.