A circle has been well defined at this point, but the circle belongs to a broader group of curves known as ellipses. We encounter ellipses quite often in nature. The orbit of all the planets around the sun follows the path of an ellipse.
The study of ellipses mathematically has many practical applications, such as in the making of convex and concave lenses for cameras, in a procedure to remove kidney stones, in telescopes, and the lenses used in our glasses, etc.
But what exactly is an ellipse? Let's look!
What is an Ellipse?
In loose terms, an ellipse is an elongated circle. In precise terms, it is defined as follows:
An ellipse is the set of points such that the sum of the distance from two fixed points (known as the foci) remains the same.
Let us mathematically outline what an ellipse is. Suppose there are two collinear points, A and B, as seen in the diagram below. We are looking for all the points such that the sum of its distance from A and B remains constant. Let P represent an arbitrary point such that
In an ellipse, the sum of distances AP and BP is constant, StudySmarter Originals
The set of all points that satisfy this condition form an ellipse.
There are a couple of terms we need to be familiar with before we get into the details of ellipses. First, there are other two points known as the vertices of the ellipse. Vertices are the endpoints of the ellipses, it is easy to visualize these vertices from the above diagram. The line on which points A and B, the center, and the vertices lie, is known as the Major Axis of the Ellipse. It is the axis along which the ellipse is aligned. The length of the major axis is measured along the two vertices i.e., the distance between the two vertices.
The Foci of an Ellipse
You may wonder what is so special about these two points, A and B. These are known as the ellipse's two foci.
The two points (A and B in the diagram) for which the sum of distances from these points is constant, which form the ellipse, are known as the foci of an ellipse.
Another property of the foci is that both foci are equidistant from the center of the ellipse. Let’s see what a center is:
The Center of an ellipse is the point where the major and minor axes of an ellipse intersect.
Alternatively, it is the point that bisects both the axes.
An ellipse looks like follows:
The center of an ellipse is where the major and minor axis intersects, StudySmarter Originals
The two points discussed earlier are the two foci A and B as shown.
The Formulas and Equations of an Ellipse
So far, we have established the idea of what an ellipse is. Now, using its properties as discussed, we will derive the equation of an ellipse. There are two types of ellipses, one which has the x-axis as its major axis, and others that have the y-axis as its major axis. Recall that d1 + d2 = 2a, where d1 and d2 are the distances of any point on the ellipse and the two foci. Let the coordinates of the two foci, A and B, be (-c,0) and (c,0), respectively. The distances AP and PB can now be calculated using the distance formula for two points on a plane.
The Equation of a Horizontal Ellipse
In a horizontal ellipse, the major axis is usually the x-axis, StudySmarter Originals
Let the coordinates of the point, as P represents any arbitrary point on the ellipse, where x and y are the variables. Using the distance formula, we have
Now adding the above two equations we get,
The derivation of the ellipse equation from this step requires a bit of algebraic manipulation. Here's how it's done!
Slightly rearranging a bit and squaring both sides gives us
Expand :
Canceling out common terms on either side of the equation highlighted above:
Now, let's isolate the radical:
Combining the cx terms, then squaring both sides again gives us:
Canceling out the common terms on the left and right, then dividing both sides by 16:
Let's put all the x's and y's on the left side of the equation, and everything else on the right:
Let's do some factoring to get
Now, let's pause the derivation for a bit and take a step back. If we denote , then using the distance formula,
And finally:
Lets sub this into the equation above!
Finally, since the final equation we want has 1 on the right side, let's divide both sides by :
There you have it!
So, we've seen how the equation of a horizontal ellipse works out to be
By setting , we obtain , which gives as the vertices of the ellipse i.e., the coordinates of the endpoints of the ellipse. Similarly, by setting , we see that the ellipse intersects the ellipse on the y-axis at points and .
An interesting property of the ellipse is that if we replace x with and y with , the equation of the ellipse remains unchanged. This implies that the ellipse remains symmetric to both the axes no matter the sign of the variable.
The Equation of a Vertical Ellipse
In a vertical ellipse, the major axis is usually the y-axis, StudySmarter Originals
The equation of the ellipse that we derived earlier is the ellipse whose major axis is the x-axis. The equation of the ellipse whose major axis is the y-axis has the same equation but with the roles of a and b reversed. The equation becomes
for which the foci lies at and vertices lie at .
Remember that the ellipses we have talked about so far have their centers aligned along the origin, and so they are called standard ellipses.
The Eccentricity of an Ellipse
Eccentricity is a quantity that has been defined to differentiate between different types of conics. It is not a geometric quantity that can be penned on a cartesian plane.
The eccentricity of a conic is the ratio of the distance of a point from its focus to the distance of the same point from its vertex. In other words, it is the measure of how oval the ellipse is.
The eccentricity of an ellipse can be found by dividing c by a, StudySmarter Originals
For convenience, the point referred to in the definition from which the distance is measured is taken to be the origin.
For an ellipse, recall that and so the value of e always lies between 0 and 1. The closer e is to 0, the more circular is the ellipse.
The ellipse has a very peculiar reflective property. It is often used to design whispering galleries where the roofs are elliptical so when a person is standing at one of the foci speaks anything, anyone on the other foci can listen. Paris subway tunnels are also created in a similar fashion where the ceilings are elliptical. The platforms are designed in such a way that the passengers on each platform are in one focus and so if one wants to say anything, anyone on the other platform can listen clearly.
Ellipses Examples
Let's look at some examples using ellipses!
Find an equation of the ellipse with vertices and foci at
Solution:
It can be seen that the foci and vertices lie on the y-axis and so the ellipse has its major axis along y-axis.
Here, and , hence
Substituting the values for a and c, we get
Therefore, the standard form of this ellipse is
Or
Find the eccentricity of an ellipse which is given by
Solution:
Comparing the equation of the given ellipse with the standard form, it is clear that and
Taking square root, we get
and as
Substituting for a and b, we get
Taking square root on both sides,
For eccentricity,
We can also make a check in order to ensure if the answer makes sense or not. We know that the eccentricity of an ellipse is between 0 and 1 so if our answer is anything beyond this range then the answer is wrong.
Find the length of the major axis and the minor axis of an ellipse whose equation is given by .
Solution:
We need to convert the given equation similar to the form below:
Where the length of the major and minor axes are 2a and 2b, respectively.
Hence, the given equation becomes:
Taking 9 in the denominator of x2 and 4 in the denominator of y2:
Which, upon further simplification, is comparable to the general form:
And when we compare both forms, we get .
Therefore the length of the major axis is and that of minor axis is .
Ellipses - Key takeaways
- An ellipse is the set of points whose sum of the distance from two fixed points is the same.
- The two fixed points from which the distance is measured are known as the ellipse's foci.
- The endpoints of an ellipse are known as the vertices of an ellipse.
- The axis on which the foci of the ellipse lie is known as the major axis of the ellipse and the perpendicular axis is known as the minor axis.
- If a is the length of the semi-major axis and b is the length of the semi-minor axis, then the distance of the foci from the origin is given by c,.
- The standard equation of an ellipse with the above defined quantities, is .
- The eccentricity of an ellipse is the measure of how oval the ellipse is, it given by and the value of e always lies between 0 and 1 for an ellipse.
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