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Question: How do I find the major and minor axes of an ellipse\(?\)...

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

Explanation

Solution

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)(a>b)
⇒(x−h)2a2+(y−k)2b2=1\Rightarrow \dfrac{{{(x-h)}^{2}}}{{{a}^{2}}}+\dfrac{{{(y-k)}^{2}}}{{{b}^{2}}}=1
And if the ellipse is vertical when (b>a)(b>a)then it will be written as
⇒(y−k)2b2+(x−h)2a2=1\Rightarrow \dfrac{{{(y-k)}^{2}}}{{{b}^{2}}}+\dfrac{{{(x-h)}^{2}}}{{{a}^{2}}}=1
In the above term (h,k)(h,k)is the center coordinate of the ellipse.
aaand bbare 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 2a2a is the length of the major axis. (h±a,k)(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±b)(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. 2b2b is the length of the minor axis. (h,k±b)(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±a,h)(k\pm a,h).
For example: If the ellipse is (x+1)24+(y−2)21=1\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>ba>b
Thus, a=2a=2 and b=1b=1 the coordinate will be (−1,2)(-1,2).

Hence the major axis length is 22 and the length of the minor axis is 11.

Note:
The eccentricity value of the ellipse is greater than or equal to zero and less than one and it is denoted by ee. If the shortest width ′b′'b' and longest width ′a′'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.