A rectangular board is 8 ft by 14 ft How far from the center

A rectangular board is 8 ft by 14 ft. How far from the center of the board should the foci be located in order to cut the largest elliptical tabletop from the board?

Answer: 5.7ft

Solution

x = 8
y = 14
Length of Major Axis = a
Length of Minor Axis = b

Center (0, 0):

h = 0
k = 0

a = 1/2 x
a = 1/2 *(14)
a = 7
b = 1/2 y
b =1/2 (8)
b = 4
So, Equation of Ellipse:

General : (x - h)² / a² + (y - k)² / b² = 1

(x - 0)² / (7)² + (y - 0)² / (4)² = 1

x² / 49 + y² / 16 = 1


Focus = c:
        _____
c = a² - b²
        _______
c = (7)² - (4)²
        ______
c = 49 - 16 = sqrt33 = 5.74

Focus : ( -5.74 , 0) and ( 5.74, 0)