Ellipse Formula

In the previous topics, we discussed the circles and their equations. Circles are really very important and popular geometry. But, the more general geometrical shape is the ellipse. Also, we will see that the circles are in particular a type of ellipse. In this topic, we will discuss the ellipses and their standard equations. The student will see the ellipse formula with some examples. Let us begin learning!

Ellipse Formula

What is Ellipse?

In geometry, we can describe an ellipse as a curve on a plane that surrounds two focal points. It is in such a way that the sum of the distances to the two focal points is constant for every point on the curve. Generally, the two focal points of the ellipse are represented as \(F_1\) and \(F_2\) and together are called as the foci of the ellipse.

Ellipse has two types of the axis – Major Axis and Minor Axis. The longest chord of the ellipse is the major axis. The perpendicular chord to the major axis is the minor axis which bisects the major axis at the center.

Here is the standard form of an ellipse.

\(\frac{{{{\left( {x – h} \right)}^2}}}{{{a^2}}}\) + \(\frac{{{{\left( {y – k} \right)}^2}}}{{{b^2}}} = 1 \)

Remember that the right side must be a 1 in order to be in standard form. Also, the point (h,k) is called as the centre of the ellipse.

To draw the graph of the ellipse all that we need are the rightmost, leftmost, topmost and bottom-most points. Once we have all these, then we can sketch in the ellipse. Here are formulas for finding these points.

(h + a,k) ,(h – a,k) ,( h,k + b ) \; and\;( h,k – b )

Note that here ‘a’ is the square root of the number under the term X. It is the amount that we move right and left from the center. Also, b is the square root of the number under the term Y. It is the amount that we move up or down from the center.

Finally, let us address the comment made at the start of section. We said that the circles are really nothing more than a special case of an ellipse. In this case we have,

\(\frac{(x – h)^2}{a^2}\)  +  \(\frac{(y – k)^2}{a^2}\) = 1

Note that we acknowledged that a=b and used a in both cases. Now if we clear denominators we get,

\({\left ( {x – h} \right)^2} + {\left ( {y – k} \right)^2} = {a^2}\)

This is the standard form of a circle with centre (h,k) and radius a. So, circles really are special cases of ellipses.

Ellipse Formula

\(\large Area\;of\;the\;Ellipse=\pi r_{1}r_{2}\)

\(\large Perimeter\;of\;the\;Ellipse=2\pi \sqrt{\frac{r_{1}^{2}+r_{2}^{2}}{2}}\)

Where,

\(r_1\) is the semi major axis for the ellipse.

\(r_2\) is the semi minor axis for the ellipse.

Solved Examples

Q.1: Find the area and perimeter of an ellipse whose semi-major axis is 12 cm and the semi-minor axis is 7 cm?

Solution:

Given in the problem,

Semi major axis of the ellipse = \(r_1\) = 12 cm

Semi minor axis of the ellipse = \(r_2\) = 7 cm

\(\large Area\;of\;the\;Ellipse=\pi r_{1}r_{2} \)

\(\large Area\;of\;the\;Ellipse=\pi \times 12 \times 7 \)

= 264 cm²

Now,

\(\large Perimeter\;of\;the\;Ellipse=2\pi \)

\(\sqrt{\frac{r_{1}^{2}+r_{2}^{2}}{2}} \)

\(\large Perimeter\;of\;the\;Ellipse=2 \times \frac{22}{7} \)

\(\sqrt{\frac{12^{2}+7^{2}}{2}} \)

= 61.74 cm