Write an equation of an ellipse in standard form with the center at the origin and with the given characteristics:
Vertex (-3,0) and co-vertex (0,2)
You have vertex at a horizontal distance of 3, so a=3
you have co-vertex at a vertical distance of 2, so b=2
Thus, your equation is
x^2/9 + y^2/4 = 1
X^2/9+y^2/4=1
To write the equation of an ellipse in standard form with the center at the origin, we need to identify the distances of the vertices and co-vertices. The standard form of an ellipse centered at the origin is:
x²/a² + y²/b² = 1
where "a" represents the distance from the center to the vertices and "b" represents the distance from the center to the co-vertices.
Given that the vertex is (-3, 0) and the co-vertex is (0, 2), we can determine the values of "a" and "b".
Distance from the center to the vertex (a) = |-3 - 0| = 3
Distance from the center to the co-vertex (b) = |0 - 2| = 2
So, the equation of the ellipse in standard form with the given characteristics is:
x²/3² + y²/2² = 1
Simplifying, the final equation is:
x²/9 + y²/4 = 1
To write the equation of an ellipse in standard form with the center at the origin, we can use the following equation:
(x-h)^2/a^2 + (y-k)^2/b^2 = 1
where (h,k) represents the coordinates of the center of the ellipse, and 'a' and 'b' represent the lengths of the major and minor axes, respectively.
In this case, the center of the ellipse is at the origin, so h = 0 and k = 0. The vertex given is (-3,0), which lies on the major axis. Since the co-vertex is (0,2), which lies on the minor axis, we can determine that the major axis length is 2 times the distance between the vertex and the center, and the minor axis length is 2 times the distance between the co-vertex and the center.
The distance between the vertex and the center is 3, so the major axis length is 2 * 3 = 6. The distance between the co-vertex and the center is 2, so the minor axis length is 2 * 2 = 4.
Now we have all the required information to write the equation:
(x-0)^2/6^2 + (y-0)^2/4^2 = 1
Simplifying further:
x^2/36 + y^2/16 = 1
Therefore, the equation of the ellipse in standard form with the given characteristics is x^2/36 + y^2/16 = 1.