Question

How do I find the major and minor axes of an ellipse$?$

Answer 1

Hint: An ellipse is a closed curve and conic section, formed by the intersection of a plane with a right circular cone. and the standard form equation of the ellipse is written as-
If the ellipse is horizontal when $(a>b)$
$\Rightarrow \dfrac{{{(x-h)}^{2}}}{{{a}^{2}}}+\dfrac{{{(y-k)}^{2}}}{{{b}^{2}}}=1$
And if the ellipse is vertical when $(b>a)$then it will be written as
$\Rightarrow \dfrac{{{(y-k)}^{2}}}{{{b}^{2}}}+\dfrac{{{(x-h)}^{2}}}{{{a}^{2}}}=1$
In the above term $(h,k)$is the center coordinate of the ellipse.
$a$and $b$are the endpoint major and minor axis.

Complete step by step solution:

The ellipse has two axes one axis is called the major axis and the other one is the minor axis
Major axis: - The longest width across the ellipse is called the major axis. For a horizontally oriented ellipse, the major axis is parallel to the x-axis. And $2a$ is the length of the major axis. $(h\pm a,k)$are the coordinates of the major axis.
For a vertically oriented ellipse, the major axis is parallel to the y-axis and the coordinates of the major axis are $(k,h\pm b)$.
Minor axis: - The shortest width across the ellipse is called the minor axis. For a horizontally oriented ellipse, the minor axis is parallel to the y-axis. $2b$ is the length of the minor axis. $(h,k\pm b)$are the coordinates of the minor axis.
For a vertically oriented ellipse, the minor axis is parallel to the x-axis and the coordinates of the minor axis are $(k\pm a,h)$.
For example: If the ellipse is $\dfrac{{{(x+1)}^{2}}}{4}+\dfrac{{{(y-2)}^{2}}}{1}=1$
Then we can see that the major axis is parallel to the x-axis because $a>b$
Thus, $a=2$ and $b=1$ the coordinate will be $(-1,2)$.

Hence the major axis length is $2$ and the length of the minor axis is $1$.

Note:
The eccentricity value of the ellipse is greater than or equal to zero and less than one and it is denoted by $e$. If the shortest width $'b'$ and longest width $'a'$ of the ellipse will be equal then it is called the circle. The two points in the ellipse are called the foci of the ellipse.