Write an equation of an ellipse with center at (0, 0), co-vertex at (-4 , 0), and focus at (0, 3).
I think its... X^2/16+y^2/9=1
To explain how to find the equation of an ellipse with the given information, let's break it down step by step:
1. Determine the center of the ellipse: The center of the ellipse is given as (0, 0). This will be important in determining the equation.
2. Find the distance between the center and the co-vertex: The co-vertex is given as (-4, 0). The distance between the center of the ellipse and the co-vertex is the value of "a" in the equation, which represents the half of the major axis. In this case, a = 4.
3. Find the distance between the center and the focus: The focus is given as (0, 3). The distance between the center of the ellipse and the focus is the value of "c" in the equation, which represents the distance from the center to one of the foci. In this case, c = 3.
4. Determine the value of "b": The value of "b" can be found using the following relationship: b^2 = a^2 - c^2. Substituting the known values, we get b^2 = 4^2 - 3^2 = 16 - 9 = 7. Taking the square root of both sides, we have b = √7.
5. Write the equation for the ellipse: The standard equation for an ellipse with its center at (h, k), and where a, b represent the half-lengths of the major and minor axes respectively, is given by ((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1.
Substituting the known values, the equation becomes ((x - 0)^2 / 4^2) + ((y - 0)^2 / √7^2) = 1.
Simplifying further, we get x^2/16 + y^2/7 = 1.
Hence, the equation of the ellipse with the given parameters is x^2/16 + y^2/7 = 1.