Applications of Conic Sections

Introduction

Conic sections have several applications in both pure and applied mathematics. We’ll go over a few of them here. When the surface of a cone connects with a plane, conic sections are generated, and they have certain features.

What are Conic Sections? 

A ‘conic section’ is the curve created by crossing a right circular cone with a plane. Euclidean geometry possesses unique features. The vertex of the cone separates it into two nappes, the upper nappe and the lower nappe.

Different types of conic sections are obtained depending on the plane’s position that intersects the cone and the angle of intersection. Specifically;

• Circle
• Ellipse
• Parabola
• Hyperbola

Curves can be found in your car’s rear view mirror or the enormous circular silver ones found at metro stations. Curves are used extensively in various fields, including the research of planetary motion, the design of telescopes, satellites, reflectors, and so on. Curves formed by the intersection of a plane with a double-napped right circular cone are known as conics. Conic sections have been discussed extensively in class 11.

Let’s look at how the different sections of the cone are formed and the formulas and their relevance.

Focus, Eccentricity and Directrix of Conic

The locus of a point P moving on the plane of a fixed-point F known as focus (F) and a fixed line d known as directrix. 

Now,

If eccentricity, e is equal to 0, the conic is a circle.

If 0<e<1, the conic is an ellipse.

If e is equal to 1, the conic is a parabola.

If e>1, it is a hyperbola

The eccentricity of an ellipse is a degree of how much it deviates from being circular. If the angle formed between the cone’s surface and its axis is β , and the angle formed between the cutting plane and the axis is  α, then the cone’s eccentricity is 

e = cos α/cos β

Important Parameters of Conic

A few other factors are described under conic sections besides focus, eccentricity, and directrix.

• Principal Axis: The line that connects the two foci or focal points of an ellipse or hyperbola. The curve’s centre is halfway.
• Linear Eccentricity: The distance between a section’s focus and its centre.
• Latus Rectum: A ratio with a focus and a parallel portion to the directrix.
• Focal Parameter: The distance between the point of focus and the directrix it corresponds to.
• Major Axis: The major axis is a chord that connects the two vertices. It is an ellipse’s longest chord.
• Minor Axis: An ellipse’s shortest chord.

Applications of Conics 

Here are the applications of conics you should know: 

1. Parabola

The use of parabolas as light or radio wave reflectors and receivers is one of the most interesting applications. Cross-sections of car headlights and flashlights, for example, are parabolas created by the paraboloid of revolution about its axis.

The arches on the river bridge in Godavari, Andhra Pradesh, India, and the Eiffel tower in Paris, France, are examples of parabolic arches that are both stable and beautiful.

2. Ellipse

All planets in the solar system, according to Johannes Kepler, rotate around the Sun in elliptic orbits with the Sun at one of the foci. For example, Halley’s Comet, which is visible once every 75 years and orbits on an elliptic orbit with an e » 0.97, is visible once every 75 years.

Our satellite moon orbits the Earth in an elliptical orbit, with one of its foci being the Earth. Satellites of other planets follow elliptical orbits around their home planets.

3. Hyperbola

Some comets travel on hyperbolic routes with the Sun at one point; unlike comets in elliptical orbits, which return at regular intervals, hyperbolic comets only pass by the Sun once.

4. Reflective property of a parabola

The parabola is a set of plane points equidistant from a fixed line (the directrix) and a fixed point (the focus) that is not on the line. As a result, the distance between a point and the focus must be equal to the distance between the point and the directrix for the point to be on a parabola.

5. Reflective Property of an Ellipse

The tangent line at a point on an ellipse forms an equal angle with the lines from the foci to that point.

Different Types of Conic Sections

Here are the different types of conic sections: 

Conclusion 

Now that you have all of the necessary information about the applications of conic sections, conic sections calculator, and more.

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