Foci of an Ellipse

Foci of Ellipse Definition

The two questions here are: What are foci? And What are ellipse? Now, first thing first, Foci are basically more than 1 focus i.e., the plural form of focus. It is pronounced as “foe-sigh”. Coming to what is an ellipse? A curved line that forms a closed-loop is known as an ellipse. The sum of the distance between foci of ellipse to any point on the line will be constant. Also, the foci are always on the longest axis and are equally spaced from the center of an ellipse. If the major and the minor axis have the same length then it is a circle and both the foci will be at the center.  


What is an Ellipse?

Most of us are well aware that our planet, the earth moves around the sun in an orbit. However, what we are not aware of is the fact that the path around which the earth moves around the sun is in the shape of a mathematical curve known as an ellipse. An ellipse almost looks like a circle but it is slightly squashed into an oval. It is like a line that can be bent around until its two ends meet, just like a circle. Things that have a shape like an ellipse are called 'elliptical'. Each ellipse has two focuses.

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Ancient Greeks while studying the conic sections, discovered for the first time an Ellipses. They obtained the ellipse by slicing a right cone at different angles. Here is a picture of a right cone being sliced at a different angle. 

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The Greeks also studied two more types of slicing of a conical structure named as hyperbola and parabola

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Properties of an Ellipse

Center

It is the point that is inside an ellipse. It is also the midpoint of the line segment that links the two foci i.e., the intersection of the major axes and the minor axes.

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Major/Minor axis

The longest diameter of an ellipse is known as the major axis and the shortest diameters of an ellipse are known as the minor axis. The length of the major axis is always equal to the sum of two generator lines. 

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Semi-major / Semi-minor Axis

The distance from the center to the furthest point on the ellipse is called semi-major and the distance from the center to the closest point on the ellipse is known as the semi-minor. They are half of the major and the minor axis respectively.

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Foci (Focus Points)

Foci are the two points on the ellipse.

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Perimeter (Circumference)

The distance around the ellipse is called the perimeter. It is slightly difficult to calculate it. 


Area

The area of an ellipse can be defined as the total number of square units that it takes to fill up the region inside an ellipse.


Chord

A chord is a line segment that links any two points on an ellipse without passing through the center.

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Tangent

A line that passes an ellipse by touching it at just one point is called a tangent.

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Secant

A secant is a line that intersects an ellipse at two points.

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How to Draw an Ellipse?

Anyone can draw an ellipse using a piece of cardboard, two soft board pins, a pencil, and a string. First, we have to fix the soft board pins in the cardboard as the foci of an ellipse. Secondly, we have to cut a piece of string but make sure that it is longer than the distance between the two soft board pins (the length of the string will be a representation of the constant in the definition). Then just tack each end of the string to the cardboard, and outline a curve with the help of a pencil held firm against the string. The result will be an ellipse.

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Calculation of the location of Foci

Now that we already know what foci are and the major and the minor axis, the location of the foci can be calculated using a formula. One thing that we have to keep in mind is that the length of the major and the minor axis forms the width and the height of an ellipse. The formula is:

F = \[\sqrt{j^{2} - n^{2}}\]

Where, F = the distance between the foci and the center of an ellipse

j = semi-major axis

n = semi-minor axis

Solved Examples

Example 1) Find the coordinates of foci using the formula when the major axis is 5 and the minor axis is 3.

Solution 1) Using the formula F = \[\sqrt{j^{2} - n^{2}}\]

F = \[\sqrt{5^{2} - 3^{2}}\]

F = \[\sqrt{25-9}\]

F = \[\sqrt{16}\]

F = 4

Foci = (0,4) & (0,-4)

Example 2) Find the coordinates of foci using the formula when the major axis is 10 and the minor axis is 6

Solution 2)  Using the formula F = \[\sqrt{j^{2} - n^{2}}\]

F = \[\sqrt{10^{2} - 6^{2}}\]

F = \[\sqrt{100 - 36}\]

F = \[\sqrt{64}\]

F = \[\sqrt{8}\]

Foci = (0,8) & (0,-8)

FAQ (Frequently Asked Questions)

Question 1. What is a Parabola and What is its Application?

Answer. A parabola can be defined as a curve that is obtained when a cone is sliced parallel to its side. The line that passes through the focus and is perpendicular to the directrix is known as “axis of symmetry”. When this axis of symmetry intersects the parabola at a particular point, we call it a vertex. The equation for a parabola is y2=x. A parabola we used to design the path of the missiles. It is also used for the headlight reflector of an automobile. Telescope, radar receiver, and satellite dishes are also examples of the application of a parabola? 

Question 2. What is a Hyperbola? 

Answer. Hyperbola can be defined as a curve that is obtained when a cone is sliced parallel to the axis. Any two hyperbola may not always be identical because of the existence of many angles between the axis and the plane. The points on the arm that are the closest are known as vertices and the line segment connecting the arm is known as the major axis. Unlike in a parabola, the two arm does not turn parallel in a hyperbola. The midpoint of the major axis is the center of the hyperbola. The equation for a hyperbola is XY=1