If the major axis is vertical and has a length of 10 units,the minor axis has a length of 8 units,and the center C=(4,-2).Fill in the missing denominators for the equation and determine the distance from C to the foci (c).
(x-4)^2/100 + (y+2)^2/64 = 1
a^2 = b^2 + c^2
c = 6
To determine the missing denominators in the equation and the distance from the center (C) to the foci (c), we can use the formula for the equation of an ellipse with a vertical major axis:
(x - h)^2
-----------
a^2 + (y - k)^2
b^2 = 1
Where (h, k) represents the center of the ellipse, a is the length of the semi-major axis, and b is the length of the semi-minor axis.
Given that the major axis length is 10 units and the minor axis length is 8 units, we can determine the values of a and b.
The length of the semi-major axis (a) is half the length of the major axis, so a = 10/2 = 5.
The length of the semi-minor axis (b) is half the length of the minor axis, so b = 8/2 = 4.
Therefore, the equation of the ellipse becomes:
(x - 4)^2 (y + 2)^2
--------------------- -------------- = 1
5^2 4^2
Simplifying further:
(x - 4)^2 (y + 2)^2
--------------------- -------------- = 1
25 16
The denominators are 25 and 16.
To determine the distance from the center (C) to the foci (c), we can use the relation:
c = sqrt(a^2 - b^2)
Substituting the values of a and b:
c = sqrt(5^2 - 4^2)
c = sqrt(25 - 16)
c = sqrt(9)
c = 3
Therefore, the distance from the center (C) to the foci (c) is 3 units.