An ellipse has the points (3, 2) and (7, 2) as foci. If the ellipse touches the x-axis, then its eccentricity is:
A. 1/\sqrt{2}
B. 1
C. 1/(2\sqrt{2})
D. \sqrt{2} - 1
Answer (1)
To solve this problem, we need to determine the eccentricity of an ellipse given certain properties.
The foci of the ellipse are given as (3, 2) and (7, 2), indicating that the major axis is horizontal since the y-coordinates are identical.
The center of the ellipse is the midpoint of the line segment joining the foci. Thus, the center (h, k) can be found by:
h = 2 3 + 7 = 5 , k = 2 2 + 2 = 2
So, the center is at (5, 2).
Since the ellipse touches the x-axis, the distance from the center to the x-axis is equal to the semi-minor axis b . In this case, since the y-coordinate of the center is 2, b = 2 .
The distance between the foci 2 c is:
c = 2 7 − 3 = 2
The relationship between the semi-major axis a , semi-minor axis b , and the distance between the foci c is given by:
a 2 = b 2 + c 2
Substituting the known values:
a 2 = 2 2 + 2 2 = 4 + 4 = 8
a = 8 = 2 2
The eccentricity e is defined as the ratio of c to a :
e = a c = 2 2 2 = 2 1
Thus, the eccentricity of the ellipse is 2 1 .
The correct choice is option A.
