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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 $\quad$ 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:

with 0 < $e$ < 1.

Because then .

**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

**Answer:**

$x^2+4y^2=16$

$\iff$ $x^2/16+y^2/4$ 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!

**Answer:**

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:

So, the ellipse equation we are looking for is:

*Question 1*:

Determine the major axis, minor axis, focal point coordinates, and eccentricity of the following ellipse equations:

**a.**

**b.**

**c.**

*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.