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
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)
FAQs on Foci of an Ellipse
1. What exactly are the foci of an ellipse as per the Class 11 syllabus?
In geometry, the foci (plural of focus) of an ellipse are two fixed points located inside the ellipse on its major axis. They are fundamental to its definition: an ellipse is the set of all points in a plane where the sum of the distances from any point on the curve to the two foci is constant. This constant sum is equal to the length of the major axis (2a).
2. What is the formula to find the coordinates of the foci of an ellipse?
The formula to find the coordinates of the foci depends on whether the ellipse is horizontal or vertical. In both cases, you first need to find the value of 'c' using the relationship c² = a² - b², where 'a' is the semi-major axis and 'b' is the semi-minor axis.
- For a horizontal ellipse (x²/a² + y²/b² = 1), the foci are located at (±c, 0).
- For a vertical ellipse (x²/b² + y²/a² = 1), the foci are located at (0, ±c).
3. How can you find the foci for a given ellipse equation, for example, x²/25 + y²/9 = 1?
To find the foci for the ellipse x²/25 + y²/9 = 1, follow these steps:
1. Identify a² and b²: Here, a² = 25 (so a = 5) and b² = 9 (so b = 3). Since a > b, the major axis is horizontal.
2. Calculate c: Use the formula c² = a² - b². This gives c² = 25 - 9 = 16. Therefore, c = 4.
3. Determine the coordinates: Since it is a horizontal ellipse, the foci are at (±c, 0). Thus, the foci for this ellipse are at (-4, 0) and (4, 0).
4. Why does an ellipse have two foci, while a circle only has one centre?
A circle is actually a special case of an ellipse where the two foci coincide. In an ellipse, the separation of the two foci determines its 'flatness' or eccentricity. As the two foci get closer to each other, the ellipse becomes more rounded. When the distance between the foci becomes zero (i.e., they merge into a single point), the ellipse becomes a perfect circle, and that single point is known as its centre.
5. How is the eccentricity of an ellipse connected to the position of its foci?
The eccentricity (e) of an ellipse is a measure of how much it deviates from being circular, and it is directly linked to the foci through the formula e = c/a. This relationship dictates the position of the foci:
- If the eccentricity e = 0, then c = 0. This means the foci are at the centre, and the ellipse is a circle.
- As the eccentricity e approaches 1, c approaches a. This means the foci move farther from the centre towards the vertices, making the ellipse more elongated and flattened.
6. What is the real-world importance of the foci of an ellipse?
The foci have a unique reflective property that is crucial in science and engineering. Any wave or ray originating from one focus will reflect off the ellipse and converge at the other focus. Key examples include:
- Astronomy: As per Kepler's First Law, planets travel in elliptical orbits with the Sun at one of the foci.
- Medicine: In lithotripsy, a medical device uses an elliptical reflector to generate shockwaves at one focus to break up kidney stones positioned at the other focus.
- Acoustics: In 'whispering galleries' with elliptical ceilings, a person whispering at one focus can be heard clearly by someone standing at the other focus, even across a large room.
7. Do the foci of an ellipse always have to lie on the major axis?
Yes, absolutely. By definition, the foci of an ellipse are always located on its major axis. The major axis is the longest diameter of the ellipse, passing through its centre and both vertices. The distance from the centre to each focus is measured along this axis, making their placement a defining characteristic of the ellipse's orientation and shape.
