Find an equation of an ellipse satisfying the given conditions:
Foci: (−2,0) and (2,0)
Length of major axis: 8
Thank you
To find the equation of an ellipse, we need to determine its center, the lengths of its major and minor axes, and the orientation (whether it is horizontally or vertically aligned).
Given the foci and the length of the major axis, we can first determine the center of the ellipse. The center lies on the major axis and is equidistant from the foci. In this case, the foci are (-2,0) and (2,0), so the center is the midpoint between these two points:
Center = ((-2 + 2)/2, (0 + 0)/2)
Center = (0, 0)
Next, we need to determine the length of the minor axis. For an ellipse, the distance between the two foci is equal to the length of the major axis. Since the foci are at (-2,0) and (2,0), the distance between them is 2*(-2) = 4. Therefore, the length of the minor axis is also 2 times the distance between the foci, which is 2*4 = 8.
Since the length of the major axis is given as 8, we can conclude that the ellipse is aligned horizontally.
Now we have all the necessary information to write the equation of the ellipse in standard form:
For a horizontally aligned ellipse:
((x - h)^2) / (a^2) + ((y - k)^2) / (b^2) = 1
where (h, k) represents the center of the ellipse, a is half the length of the major axis, and b is half the length of the minor axis.
Plugging in the values we know:
((x - 0)^2) / (4^2) + ((y - 0)^2) / (8^2) = 1
Simplifying the equation gives us the final result:
(x^2) / 16 + (y^2) / 64 = 1
So, the equation of the ellipse satisfying the given conditions is (x^2) / 16 + (y^2) / 64 = 1.
To find the equation of an ellipse given its foci and the length of the major axis, we can use the following formula:
c = √(a^2 - b^2)
where:
- c is the distance between each focus and the center of the ellipse
- a is the length of the major axis (which equals half of the length of the major axis)
- b is the length of the minor axis (which equals half of the length of the minor axis)
In this case, we know that the foci are (-2, 0) and (2, 0), and the length of the major axis is 8. Therefore:
- a = 8/2 = 4
- c = 2
Using the formula, we can find the value of b:
c = √(a^2 - b^2)
2 = √(4^2 - b^2)
2 = √(16 - b^2)
Squaring both sides of the equation:
4 = 16 - b^2
b^2 = 16 - 4
b^2 = 12
Now we have the values of a, b, and c, so we can write the equation of the ellipse:
(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1
where (h, k) is the center of the ellipse. Since the foci are located at (-2, 0) and (2, 0), the center is at the midpoint between these two points, which is (0, 0).
Substituting the values of a, b, and (h, k) into the equation, we get:
(x - 0)^2 / 4^2 + (y - 0)^2 / (√12)^2 = 1
x^2 / 16 + y^2 / 12 = 1
Therefore, the equation of the ellipse satisfying the given conditions is:
x^2 / 16 + y^2 / 12 = 1
in its standard form , at centre (0,0)
a^2 = b^2 + c^2
4^2 = b^2 + 2^2
b^2 = 12
x^2 / 16 + y^2 / 12 = 1