Difference Between Parabola and 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

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

$\frac{a^{2}x}{x_{1}}$ + $\frac{b^{2}y}{y_{1}}$ = a² + b²

Equation of Normal at point (6, 3) is

$\frac{18x}{6} + \frac{9y}{3} = 18 + 9$

3x + 3y = 27

3(x + y) = 27

$x + y = \frac{27}{3}$

x + y = 9

Chord of contact:

T = 0

$\frac{xx_{1}}{a^{2}} - \frac{yy_{1}}{b^{2}} = 1$

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.