How can you tell from the equation of an ellipse which is the major and minor axis?
The major and minor axes are perpendicular to one another and are two axes of symmetry. For an ellipse, one is longer than the other. The major axis is the longer one. The major axis passes through the two foci of the ellipse.
Below is a typical ellipse equation.
[(x - 6)^2]/9 + [( y + 4)^2]/25 = 1
Do you see the denominators 9 and 25?
If you take the square root of each denominator, what is the biggest and smallest result?
The square root of 9 = 3.
The square root of 25 = 5.
From this we know that the smaller number 3 represents the minor axis and 5 (the bigger number) represents the major axis.
We also know, from the example above, that since the smaller number lies under the x coordinate, the minor axis is horizontal and the major axis is vertical for this equation. If 25 was the denominator of the x coordinate, then the minor axis would be vertical and the major axis would be horizontal.
Is this clear?
To determine the major and minor axes of an ellipse from its equation, follow these steps:
1. Write down the equation of the ellipse in standard form:
(x-h)²/a² + (y-k)²/b² = 1
Here, (h, k) represents the center of the ellipse, and "a" and "b" are the semi-major and semi-minor axes, respectively.
2. Compare the coefficients of x² and y²:
The coefficient of x², which is a², represents the square of half the length of the major axis.
The coefficient of y², which is b², represents the square of half the length of the minor axis.
3. Take the square root of the coefficients:
Calculate the square root of a² to find the length of the semi-major axis.
Calculate the square root of b² to find the length of the semi-minor axis.
4. Determine which axis is longer:
Compare the lengths of the semi-major and semi-minor axes to identify which one is longer. The longer axis corresponds to the major axis of the ellipse, while the shorter axis corresponds to the minor axis.
By analyzing the equation of the ellipse and finding the square root of the coefficients, you can discern the major and minor axes of the ellipse.