Find an equation of the ellipse with its center at the origin.
center(0,0)
foci(2,0)(-2,0)
major axis=8
My answer is x^2/16 + y^2/15 = 1. Is this correct?
major axis = 8, so a = 8
foci (2,0), so c = 2
b^2 + c^2 = a^2
b^2 + 4 = 64
b^2 = 60
x^2/64 + y^2/60 = 1
To find the equation of an ellipse with its center at the origin, we can use the formula:
(x-h)^2/a^2 + (y-k)^2/b^2 = 1
where (h,k) represents the center of the ellipse, and a and b represent the semi-major and semi-minor axes, respectively.
From the given information, we know that the center of the ellipse is (0,0) and the foci are located at (2,0) and (-2,0).
The distance between the center and each focus is called the "c" value of the ellipse. In this case, c = 2.
We can calculate the value of "a" using the relationship: c = sqrt(a^2 - b^2).
Substituting the values, we have:
2 = sqrt(a^2 - b^2)
Squaring both sides of the equation, we get:
4 = a^2 - b^2
We also know that the length of the major axis is 8, which means 2a = 8. Solving for "a", we find a = 4.
Substituting this value back into the equation from earlier, we have:
4 = 16 - b^2
Rearranging the equation, we get:
b^2 = 12
Now we have the values of "a" and "b". Plugging them into the equation for the ellipse, we get:
x^2/16 + y^2/12 = 1
So, the correct equation for the given ellipse is x^2/16 + y^2/12 = 1.