Write the standard form equation for the following Ellipse. Use the information given below.
Foci: (0,0) and (4,0)
Major Axis of length 8
This must be re your previous post
from above: centre is (2,0)
c is from centre to focus or c = 2
2a = 8
a = 4
you should know that for a horizontal ellipse,
a^2 = b^2 + c^2
16 = b^2 + 4
b^2 = 12
(x-2)^2 /16 + y^2 /12 = 1
To write the standard form equation for an ellipse, we need to know the coordinates of the foci and the length of the major axis.
First, let's label the coordinates of the foci: F1 = (0,0) and F2 = (4,0).
The major axis is the line segment that passes through the foci and has its endpoints on the ellipse. We are given that the length of the major axis is 8. Since the foci are (0,0) and (4,0), this means that the length of the major axis is twice the distance between the foci: 8 = 2 * distance between foci.
Using the distance formula, we can calculate the distance between the foci:
distance between foci = sqrt((0 - 4)^2 + (0 - 0)^2)
= sqrt((-4)^2 + 0^2)
= sqrt(16)
= 4
Now that we know the distance between the foci, we can write the standard form equation for the ellipse.
For an ellipse centered at the origin (0,0), the standard form equation is:
(x^2 / a^2) + (y^2 / b^2) = 1
where "a" is the distance from the center to the vertices along the x-axis (semi-major axis), and "b" is the distance from the center to the vertices along the y-axis (semi-minor axis).
Since the foci are on the x-axis, the value of "a" is half the length of the major axis:
a = 8 / 2
= 4
And we already know the distance between the foci is 4. Since the foci are on the x-axis, this means the value of "c" (the distance from the center to each focus) is also 4.
Now we can substitute the values into the standard form equation:
(x^2 / 4^2) + (y^2 / b^2) = 1
Since the foci are on the x-axis and the major axis is parallel to the x-axis, the equation simplifies to:
(x^2 / 4^2) + (y^2 / b^2) = 1
Simplifying further, we have:
(x^2 / 16) + (y^2 / b^2) = 1
This is the standard form equation for the given ellipse. The only remaining unknown is the value of "b", which we can find using additional information about the ellipse, such as its minor axis or any other given points on the ellipse.