Conic Section, Objectives, Conic Section, Ellipse, Different Shapes

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While cutting a carrot you might have noticed different shapes shown by the edges of the cut.

Analytically you may cut it in three different ways, namely

Cut is parallel to the base (see Fig.1)

Cut is slanting but does not pass through the base (see Fig.2)

Cut is slanting and passes through the base (see Fig.3)

Cutting a carrot

Cutting a Carrot

Cutting a carrot

The different ways of cutting, give us slices of different shapes.

Objectives

After studying this lesson, you will be able to:

After studying this lesson

After Studying this Lesson

After studying this lesson

Conic Section

In the introduction we have noticed the various shapes of the slice of the carrot. Since the carrot is conical in shape so the section formed are sections of a cone. They are therefore called conic sections.

Mathematically, a conic section is the locus of a point which moves so that its distance from a fixed point is always in a constant ratio to its perpendicular distance from a fixed line.

The fixed point is called the focus and is usually denoted by .

The fixed straight line is called the Directrix.

The straight line passing through the focus and perpendicular to the Directrix is called the axis.

The constant ratio is called the eccentricity and is denoted by .

What happens when?

(ii) (iii)

In these cases the conic section obtained are known as ellipse, parabola and hyperbola respectively.

Ellipse

The slice thus obtained represents an ellipse. Let us define the ellipse mathematically as follows:

“An ellipse is the locus of a point which moves in a plane such that its distance from a fixed point bears a constant ratio to its distance from a fixed line and this ratio is less than unity”.

Developed by: