1. What is the equation an ellipse with
foci at (0, ±2 √10 ) and
x-intercept at (9, 0)?
2. What is the equation of an ellipse
with foci at (±4, 0) and a minor axis
of length 6?
To find the equation of an ellipse, we need to know the coordinates of its foci, as well as additional information such as the lengths of its major and minor axes or the lengths of its semi-major and semi-minor axes.
1. For the first question, we are given the foci at (0, ±2√10) and one of the x-intercepts at (9, 0). To find the equation of the ellipse, we need to determine the length of the major axis and the length of the minor axis.
Since the foci are located on the y-axis, the major axis of the ellipse will be vertical. The distance between the foci is the length of the major axis, which is equal to 2a.
Given that the foci are at (0, ±2√10), we have 2a = 2(2√10) = 4√10.
The x-intercept at (9, 0) will be a point on the major axis, and its distance from the center of the ellipse is half the length of the major axis. Therefore, the distance between (0, 0) and (9, 0) is a.
We can now find the value of a. Using the Pythagorean theorem, we have:
a^2 + b^2 = c^2,
where c is the distance between the center of the ellipse and one of the foci.
Since the center of the ellipse is at (0, 0) and the foci are at (0, ±2√10), the distance c is equal to 2√10. Thus, we have:
a^2 + b^2 = (2√10)^2,
a^2 + b^2 = 40.
Since the distance between the center and one end of the major axis is a = 9, we can substitute this value into the equation above to solve for b:
9^2 + b^2 = 40,
81 + b^2 = 40,
b^2 = 40 - 81 = -41.
As b^2 is negative, it means that the equation does not represent an ellipse. Instead, it represents an empty set, as a square root of a negative number is not a real number. Therefore, the given information does not provide values for a valid equation of an ellipse.
2. For the second question, we are given the foci at (±4, 0) and the length of the minor axis, which is equal to 6. To find the equation of the ellipse, we need to determine the length of the major axis and the coordinates of the center.
Given that the foci are on the x-axis, the major axis of the ellipse will be horizontal. The distance between the foci is the length of the major axis, which is equal to 2a.
Given that the foci are at (±4, 0), we have 2a = 2(4) = 8.
The length of the minor axis is 6, and the minor axis is perpendicular to the major axis. Thus, the distance between the center and one end of the minor axis is half the length of the minor axis, which is 6/2 = 3.
Since the center of the ellipse is located between the foci, the center is at (0, 0).
Using these values, we can now find the equation of the ellipse. The general form of the equation for an ellipse with a horizontal major axis is:
[(x - h)^2 / a^2] + [(y - k)^2 / b^2] = 1,
where (h, k) represents the coordinates of the center, a represents half the length of the major axis, and b represents half the length of the minor axis.
Plugging in the values, we have:
[(x - 0)^2 / (8/2)^2] + [(y - 0)^2 / (6/2)^2] = 1,
x^2 / 4^2 + y^2 / 3^2 = 1,
x^2 / 16 + y^2 / 9 = 1.
Therefore, the equation of the ellipse with foci at (±4, 0) and a minor axis of length 6 is x^2 / 16 + y^2 / 9 = 1.