Ellipse
Ellipse is an integral part of the conic section and is similar in properties to a circle. Unlike the circle, an ellipse is oval in shape. An ellipse has an eccentricity less than one, and it represents the locus of any point whose sum of the distances from the two foci of the ellipse is a constant value. A simple example of the ellipse in our daily life is the shape of an egg in a twodimensional form and the running tracking in a sports stadium.
Here we shall aim at knowing the definition of an ellipse, the derivation of the equation of an ellipse, and the different standard form of equations of the ellipse.
Ellipse  Definition
An ellipse in math is the locus of a plane point in such a way that its distance from a fixed point has a constant ratio of \(e\) to its distance from a fixed line (less than 1). The ellipse is a part of the conic segment, which is the intersection of a cone with a plane that does not intersect the cone's base. The fixed point is called the focus and is denoted by S, the constant ratio \(e\) as the eccentricity, and the fixed line is called as directrix (d) of the ellipse.
Also, an ellipse is the locus of a point, the sum of whose distances from two fixed points is a constant value. The two fixed points are called the foci of the ellipse.
\(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\)
Let us check through a few important terms relating to an ellipse.
 Focus: The ellipse has two foci and their coordinates are F(c, o), and F'(c, 0).
 Center: The midpoint of the line joining the two foci is called the center of the ellipse.
 Major Axis: The length of the major axis of the ellipse is 2a units, and the end vertices of this major axis is (a, 0), (a, 0) respectively.
 Minor Axis: The length of the minor axis of the ellipse is 2b units and the end vertices of the minor axis is (0, b), and (0, b) respectively.
 Latus Rectum: The latus rectum is a line drawn perpendicular to the transverse axis of the ellipse and is passing through the foci of the ellipse. The length of the latus rectum of the ellipse is 2b^{2}/a.
 Transverse Axis: The line passing through the two foci and the center of the ellipse is called the transverse axis.
 Conjugate Axis: The line passing through the center of the ellipse and perpendicular to the transverse axis is called the conjugate axis
 Eccentricity: (e < 1). The ratio of the distance of the focus from the center of the ellipse, and the distance of one end of the ellipse from the center of the ellipse. If the distance of the focus from the center of the ellipse is 'c' and the distance of the end of the ellipse from the center is 'a', then eccentricity e = c/a.
Derivation  Ellipse Equation
The first step in the process of deriving the equation of the ellipse is to derive the relationship between the semimajor axis, semiminor axis, and the distance of the focus from the center. The aim is to find the relationship across a, b, c. The length of the major axis of the ellipse is 2a and the length of the minor axis of the ellipse is 2b. The distance between the foci is equal to 2c. Let us take a point P at one end of the major axis and aim at finding the sum of the distances of this point from each of the foci F and F'.
PF + PF' = OP  OF + OF' + OP
= a  c + c + a
PF + PF'= 2a
Now let us take another point Q at one end of the minor axis and aim at finding the sum of the distances of this point from each of the foci F and F'.
QF + QF' = \(\sqrt{b^2 + c^2}\) + \(\sqrt{b^2 + c^2}\)
QF + QF' = 2\(\sqrt{b^2 + c^2}\)
The points P and Q lie on the ellipse, and as per the definition of the ellipse for any point on the ellipse, the sum of the distances from the two foci is a constant value.
2\(\sqrt{b^2 + c^2}\) = 2a
\(\sqrt{b^2 + c^2}\) = a
b^{2} + c^{2} = a^{2}
c^{2} = a^{2}  b^{2}
Let us now check, how to derive the equation of an ellipse. Now we consider any point S on the ellipse and take the sum of its distances from the two foci F and F', which is equal to 2a units. If we observe the above few steps, we have already proved that the sum of the distances of any point on the ellipse from the foci is equal to 2a units.
SF + SF' = 2a
\(\sqrt{(x + c)^2 + y^2}\) + \(\sqrt{(x  c)^2 + y^2}\) = 2a
\(\sqrt{(x + c)^2 + y^2}\) = 2a  \(\sqrt{(x  c)^2 + y^2}\)
Now we need to square on both sides to solve further.
(x + c)^{2} + y^{2} = 4a^{2 }+ (x  c)^{2} + y^{2}  4a\(\sqrt{(x  c)^2 + y^2}\)
x^{2} + c^{2} + 2cx + y^{2} = 4a^{2 }+ x^{2} + c^{2}  2cx + y^{2}  4a\(\sqrt{(x  c)^2 + y^2}\)
4cx  4a^{2} =  4a\(\sqrt{(x  c)^2 + y^2}\)
a^{2}  cx = a\(\sqrt{(x  c)^2 + y^2}\)
Squaring on both sides and simplifying, we have.
\(\dfrac{x^2}{a^2} + \dfrac{y^2}{c^2  a^2} =1\)
Since we have c^{2} = a^{2}  b^{2 }we can substitute this in the above equation.
\(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} =1\)
This derives the standard equation of the ellipse.
Standard Equations of an Ellipse
There are two standard equations of the ellipse. These equations are based on the transverse axis and the conjugate axis of each of the ellipse. The standard equation of the ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) has the transverse axis as the xaxis and the conjugate axis as the yaxis. Further, another standard equation of the ellipse is \(\dfrac{x^2}{b^2} + \dfrac{y^2}{a^2} = 1\) and it has the transverse axis as the yaxis and its conjugate axis as the xaxis. The below image shows the two standard forms of equations of an ellipse.
Properties of an Ellipse
The properties of an ellipse are given as,
 An ellipse is created at the angle of its base by a plane intersecting a cone.
 All ellipses have two foci or focal points. The sum of the distances from any point on the ellipse to the two focal points is a constant value.
 There is a center and a major and minor axis in all ellipses.
 The eccentricity value of all ellipses is less than one.
Let us check through three important terms relating to an ellipse.
 Auxilary Circle: A circle drawn on the major axis of the ellipse as its diameter is called the auxiliary circle. The equation of the auxiliary circle to the ellipse is x^{2} + y^{2} = a^{2}.
 Director Circle: The locus of the point of intersection of the perpendicular tangents drawn to the ellipse is called the director circle. The equation of the director circle of the ellipse is x^{2} + y^{2} = a^{2} + b^{2}
 Parametric Coordinates: The parametric coordinates of any point on the ellipse is (x, y) = (aCosθ, aSinθ). These coordinates represent all the points of the coordinate axes and it satisfies all the equations of the ellipse.
How to Draw an Ellipse?
To draw an ellipse in math, there are certain steps to be followed
 Decide what length the major axis will be because the major axis is the longest diameter of an ellipse.
 Draw one horizontal line of major axis length.
 Mark the midpoint with a ruler. This can be done by taking the length of the major axis and dividing it by two.
 Construct a circle of this diameter with a compass.
 Decide what length the minor axis will be because the minor axis is the shortest diameter of an ellipse.
 Now, at the midpoint of the major axis, you take the protractor and set its origin. At 90 degrees, mark the point. Then swing 180 degrees with the protractor and mark the spot. You may now draw the minor axis between or within the two marks at its midpoint.
 Draw a circle of this diameter with a compass as we did for the major axis.
 Use a compass to divide the entire circle into twelve 30 degree parts. Setting your protractor on the main axis at the origin and labeling the intervals of 30 degrees with dots will do this. Then with lines, you can link the dots through the middle.
 Draw horizontal lines (except for the major and minor axes) from the inner circle.
They are parallel to the main axis, and from all the points where the inner circle and 30degree lines converge, they go outward.
Try drawing the lines a little shorter near the minor axis, but draw them a little longer as you move toward the major axis.  Draw vertical lines (except for the major and minor axes) from the outer circle.
These are parallel to the small axis, and from all the points where the outer circle and 30degree lines converge, they go inward.
Try to draw the lines a little longer near the minor axis, but when you step towards the main axis, draw them a little shorter.
You can take a ruler and stretch it a little before drawing the vertical line if you detect that the horizontal line is too far.  Do your best with freehand drawing to draw the curves between the points by hand.
Plotting an Ellipse
Let us see the graphical representation of an ellipse with the help of an ellipse formula.
There are certain steps to be followed to graph ellipse.
Step 1: Intersection with the coordinate axes
The ellipse intersects the xaxis in the points A (a, 0), A'(a, 0) and the yaxis in the points B(0,b), B'(0,b).
Step 2 : The vertices of the ellipse are A(a, 0), A'(a, 0), B(0,b), B'(0,b).
Step 3 : Since the ellipse is symmetric about the coordinate axes, the ellipse has two foci S(ae, 0), S'(ae, 0) and two directories d and d' whose equations are \(x = \frac{a}{e}\) and \(x = \frac{a}{e}\). The origin O bisects every chord through it. Therefore, origin O is the centre of the ellipse. Thus it is a central conic.
Step 4: The ellipse is a closed curve lying entirely within the rectangle bounded by the four lines \(x = \pm a\) and \(y = \pm b\).
Step 5: The segment \(AA'\) of length \(2a\) is called the major axis and the segment \(BB'\) of length \(2b\) is called the minor axis. The major and minor axes together are called the principal axes of the ellipse.
The length of semimajor axis is \(a\) and semiminor axis is b.
Solved Examples on Ellipse

Example 1: What are the values of \(a\) & \(b\) in the equation of ellipse \(16x^2 + 25y^2 = 1600\).
Solution:
To get the values of \(a\) & \(b\) we need to write the given equation in standard form
So divide the given equation of an ellipse \(16x^2 + 25y^2 = 1600\) by 1600.
we get \(\frac{{x^2}}{100} + \frac{{y^2}}{64} = 1\)
So by comparing \(\frac{{x^2}}{100} + \frac{{y^2}}{64} = 1\) with \(\frac{{x^2}}{{a^2}} + \frac{{y^2}}{{b^2}} = 1\)
The values \(a=10\) and \(b=8\)

Example 2: Find the lengths of major and minor axes of the ellipse \(\frac{{x^2}}{25} + \frac{{y^2}}{16} = 1\).
Solution:
The equation of the ellipse is \(\frac{{x^2}}{25} + \frac{{y^2}}{16} = 1\)
Comparing the above equation with \(\frac{{x^2}}{a^2} + \frac{{y^2}}{b^2} = 1\),
\(a^2=25\) and \(b^2=16\)
Lenght of major axis = 2a = 10
Lenght of minor axis = 2b = 8

Example 3: Find the equation of the ellipse, having major axis along the Xaxis and passing through the points (3, 1) and (2, 2).
Solution:
Since the points (3, 1) and (2, 2) lie on the ellipse,
\(\frac{{x^2}}{a^2} + \frac{{y^2}}{b^2} = 1\), \(\frac{{9}}{a^2} + \frac{{1}}{b^2} = 1\) and
\(\frac{{4}}{a^2} + \frac{{4}}{b^2} = 1\)
Solving these equations simultaneously \(a^2 = \frac{{32}}{3}\) and \(b^2 = \frac{{32}}{5}\)
So the equation of the ellipse is
\(\frac{{x^2}}{{\frac{{32}}{3}}} + \frac{{y^2}}{{\frac{{32}}{5}}} = 1\)
i.e. \(3x^2 + 5y^2 = 32\)
FAQs on Ellipse
What is Ellipse in Conic Section?
An ellipse is the locus of a point whose sum of the distances from two fixed points is a constant value. The two fixed points are called the foci of the ellipse, and the equation of the ellipse is \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\). Here a is called the semimajor axis and b is called the semiminor axis of the ellipse.
What Is the Equation of Ellipse?
The equation of the ellipse is \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\). Here a is called the semimajor axis and b is the semiminor axis. For this equation, the origin is the center of the ellipse and the xaxis is the transverse axis, and the yaxis is the conjugate axis.
How to Find Equation of an Ellipse?
The equation of the ellipse can be derived from the basic definition of the ellipse: An ellipse is the locus of a point whose sum of the distances from two fixed points is a constant value. Let the fixed point be P(x, y), the foci are F and F'. Then the condition is PF + PF' = 2a. This on further substitutions and simplification we have the equation of the ellipse as \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\).
What is the Eccentricity of Ellipse?
The eccentricity of the ellipse refers to the measure of the curved feature of the ellipse. For an ellipse, the eccentric is always greater than one. (e < 1). Eccentricity is the ratio of the distance of the focus and one end of the ellipse, from the center of the ellipse. If the distance of the focus from the center of the ellipse is 'c' and the distance of the end of the ellipse from the center is 'a', then eccentricity e = c/a.
What Is the Foci of an Ellipse?
The ellipse has two foci, F and F'. The midpoint of the two foci of the ellipse is the center of the ellipse. All the measurements of the ellipse are with reference to these two foci of the ellipse. As per the definition of an ellipse, an ellipse includes all the points whose sum of the distances from the two foci is a constant value.
What Is the Conjugate Axis of an Ellipse?
The axis passing through the center of the ellipse, and which is perpendicular to the line joining the two foci of the ellipse is called the conjugate axis of the ellipse. For a standard ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\), its minor axis is yaxis, and it is the conjugate axis.
What are Asymptotes of Ellipse?
The ellipse does not have any asymptotes. Asymptotes are the lines drawn parallel to a curve and are assumed to meet the curve at infinity. We can draw asymptotes for a hyperbola.
What Are the Vertices of an Ellipse?
There are four vertices of the ellipse. The length of the major axis of the ellipse is 2a and the endpoints of the major axis is (a, 0), and (a, 0). The length of the minor axis of the ellipse is 2b and the endpoints of the minor axis is (0, b), and (0, b).
How to Find Transverse Axis of an Ellipse?
The line passing through the two foci and the center of the ellipse is called the transverse axis of the ellipse. The major axis of the ellipse falls on the transverse axis of the ellipse. For an ellipse having the center and the foci on the xaxis, the transverse axis is the xaxis of the coordinate system.