The Standard Equation Of An Ellipse

Ellipse is a member of the conic section and has an oval shape. Its eccentricity is less than one, and the total of their distances from the ellipse’s two foci is a constant value. The shape of a rugby ball or a blown rubber balloon is a simple example of an ellipse in our daily lives.

We will learn about the definition of an ellipse, the derivation of an ellipse’s equation, and the standard equation of an ellipse.

What is an ellipse?

An ellipse is a set of all points in a plane whose sum of distances from two fixed points in the plane is constant. When a line segment is drawn between the two focus points, the line’s midpoint is the ellipse’s centre.

The main axis connects the two foci, while the minor axis is a line drawn through the centre and perpendicular to the major axis. The vertices of the ellipse are the primary axis’ endpoints.

In addition,

‘2a’ is the length of the principal axis.

‘2b’ is the length of the minor axis.

‘2c’ is the distance between the foci.

Parts of an ellipse

Let’s go through a few keywords related to the various sections of an ellipse:

• Foci of an ellipse: F(c, o), and F’ are the coordinates of the two foci on the ellipse (-c, 0). As a result, the distance between the foci is equal to 2c.

• Centre of the ellipse: The centre of the ellipse is the middle of the line connecting the two foci.

• Transverse axis: The transverse axis is the line that passes across the two foci and the ellipse’s centre.

• Conjugate axis: The conjugate axis is a line that passes through the ellipse’s centre and is perpendicular to the transverse axis.

• The major axis of the ellipse: The ellipse’s major axis is 2a units long, and its end vertices are (a, 0), (-a, 0), respectively.

• The minor axis of the ellipse: The length of the ellipse’s minor axis is 2b units, and its end vertices are (0, b) and (0, -b), respectively.

Standard equation of an ellipse

There are two standard ellipse equations. The transverse and conjugate axes of each ellipse are used to derive these equations. The transverse axis is the x-axis, and the conjugate axis is the y-axis, in the basic elliptical equation

x2/a2 + y2/b2 = 1

 x2/b2 + y2/a2 = 1, which is another example of the standard equation of an ellipse, with the transverse axis as the y-axis and the conjugate axis as the x-axis. 

Derivation of the standard equation of an ellipse

Now, on the ellipse, find a point P(x, y) such that PF1 + PF2 = 2a. 

According to the distance formula:

√ {(x + c)2 + y2} + √ {(x – c)2 + y2} = 2a

Alternatively, √ {(x + c)2 + y2} = 2a – √ {(x – c)2 + y2}

Let’s also square both sides. As a result, we have (x + c)2 + y2 = 4a2 – 4a√ {(x – c)2 + y2} + (x – c)2 + y2

Simplifying the equation, we get √ {(x – c)2 + y2} = a – x(c/a)

We simplify it further by squaring both sides to get x2/a2 + y2/(a2 – c2) = 1.

c2 = a2 – b2 is a well-known formula. As a result, x2/a2 + y2/b2 = 1.

As a result, any point on the ellipse will satisfy the equation:

1 = x2/a2 + y2/b2.. (1)

Converse form

Let’s take a look at the scenario from the other side. y2 = b2(1 – (x2/a2)) if P(x, y) fulfils equation (1) with 0 c a.

As a result, PF1 = (x + c)2 + y2 = (x + c)2 + b2(1 – (x2/a2)) = (x + c)2 + b2(1 – (x2/a2))

By replacing b2 with a2 – c2, the equation becomes PF1 = a + x(c/a).

PF2 = a – x(c/a) is obtained using similar techniques.

As a result, PF1 + PF2 equals a + x(c/a) + a – x(c/a) = 2a.

As a result, any point on the ellipse that fulfils equation (1), i.e. x2/a2 + y2/b2 = 1, is on it. Also, an ellipse with the origin in its centre and a major axis along the x-axis has the equation:

1 = x2/a2 + y2/b2.

Solving the equation (1), we get

x2/a2 = 1 – y2/b2 ≤ 1

Therefore, x2 ≤ a2. So, – a ≤ x ≤ a. Hence, we can say that the ellipse lies between the lines x = – a and x = a and touches these lines. Similarly, it can be between the lines y = – b and y = b and touch those lines.

Hence the examples of the standard equations of an ellipse are:

x2/a2 + y2/b2 = 1.

x2/b2 + y2/a2 = 1.

Conclusion

The standard forms of equations reveal fundamental characteristics of graphs. By learning the standard equation of an ellipse, we can interpret the standard equation of ellipse questions.