ELLIPTICAL PROPERTIES - mathematic.my.id

# ELLIPTICAL PROPERTIES

In this post, we will discuss the properties of elliptical shapes. Previously what is an ellipse, well to understand it, consider the following image:
From the picture above, the properties of elliptical given as follows:
1. Elliptical has major axis (long axis) and minor axis (short fuse). Pay attention to the image above which is a major axis is $AA'$ and minor axis is a $BB'$.
2. Elliptical $\quad$$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ cut the x-axis at the point $(a, 0)$ and $(-a, 0)$ as well as cut y-axis at the point $(0, b)$ and $(0,-b)$. So that the long axis = $2a$ and the length of minor axis major = $2b$.
3. The axis of the elliptical symmetry is the axis of the major and minor axis intersect in the central point of ellipse.
4. The major axis and the minor axis intersect the ellipse at the vertex of the ellipse. In the image above which is the top of the elliptical is the point $A (a, 0)$, $A'(-a, 0)$, $B(0, b)$, and $B'(0,-b)$.
5. The ellipse has two focuses located on the major axis. For an ellipse horizontal position whose center is (0, 0) then the focus is $F_1(-c, 0)$ and $F_2(c, 0)$, while the ellipse whose center is $(p, q)$ then the focus is $F_1(p-c, q)$ and $F_2(p+c, q)$. And for an ellipse with a vertical position,
$F_1(0, -c)$ and $F_2(0, c)$ {the ellipse is centered at $O(0, 0)$}, and
$F_1(p, q-c)$, $F_2(p, q+c)$ {ellipse centered at $(p, q)$.}
6. Comparison of the distance from a point on elliptical to the focal point with direktriks lines called eccentricity, abbreviated $e$. The magnitude of the eccentricity $(e)$ is an:
$e=\frac{c}{a}$ with 0 < $e$ < 1.
Because $c=\sqrt{a^{2}-b^{2}}$ then $c=\frac{\sqrt{a^{2}-b^{2}}}{a}$.

Example 1:
You know the ellipse with the equation $x^2+4y^2=16$. Tentukan:
a. Major axis
b. Minor axis
c. Focal point coordinates
d. Eccentricity
$x^2+4y^2=16$
$\iff$ $x^2/16+y^2/4$ diperoleh $a^2=16$ atau $a=4$ dan $b^2=4$ atau $b=2$ and
$c=\sqrt{a^2-b^2}=\sqrt{16-4}=2\sqrt{3}$.
a. Major axis = $2a=2(4)=8$.
b. Minor axis = $2b=2(2)=4$.
c. Focal point coordinates is $F_1(-2\sqrt{3}, 0)$ dan $F_2(2\sqrt{3}, 0)$.
d. Eccentricity $e=c/a=(2\sqrt{3})/4=\sqrt{3}/2$.

Example 2:
Find the equation for an ellipse centered at $O(0, 0)$, where one of the foci is on point $(0, \sqrt{5})$ and the axis is 6 units long!
in the above problem it is clear that the ellipse formed is an ellipse with a vertical position. From property 5, it is obtained $c=\sqrt{5}$,
The long axis = $2a = 6$ then $a = 3$, and $b = \sqrt{a^2-c^2}=\sqrt{9-5}=2$.
since the ellipse is vertical, the equation is:
$\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$
So, the ellipse equation we are looking for is: $\frac{x^2}{4}+\frac{y^2}{9}=1$

Question 1:
Determine the major axis, minor axis, focal point coordinates, and eccentricity of the following ellipse equations:
a. $\frac{x^{2}}{169}+\frac{y^{2}}{25}=1$
b. $4x^{2}+2y^{2}=8$
c. $x^{2}+4y^{2}=4$
Question 2:
Find the equation of an ellipse centered at (0, 0), one of its foci on $(\sqrt{3}, 0)$ with a major axis of 4 units!

Thus a brief explanation of the properties of an ellipse, good-bye and hopefully useful.