Write answers as decimal numbers rounded to the nearest hundredth (when necessary). If the equation describes a circle, state that and find the diameter and eccentricity.

Question

For each of the following ellipses, determine:

a) the direction of the major axis

b) the length of the major axis

c) the length of the minor axis

d) the eccentricity of the ellipse

1. (x-3)²/25 + (y-2)²/9 = 1
2. (x-5)²/4 + (y+3)²/49 = 1

Answer 1

(x-h)^/a^2 + (y-k)^2/b^2 = 1 is the general equation for an ellipse where 2a is the major axis, 2b is the minor axis when a>b, eccentricity = c/a where c=sqr(a^2-b^2)

In the first equation a=5, b=3, c=4, the major axis is horizontal =10, minor axis =6

eccentricity = 4/5 = 0.8

second equation is another ellipse. Neither is a square unless a=b major axis is vertical, 2a =14 = major axis 2b=4 = minor axis

c=sqr(49-4) e=c/a = sqr(45)/7 = (3/7)sqr5 = about 0.96

Answer 2

1. (x-3)²/25 + (y-2)²/9 = 1

a = 5, b = 3, center is (3, 2)

a) Major axis II x-axis and belong to line y = 2

b) 2a = 10

c) 2b = 6

d) c² = 25 - 9 = 16; c = 4 and eccentricity e = c/a = 4/5 = 0.8

2.(x-5)²/4 + (y+3)²/49 = 1

a = 7, b = 2, center is (5, -3)

a) Major axis II y-axis and belong to line x = 5

b) 2a = 14

c) 2b = 4

d) Eccentricity e = c/a; c² = 49 - 4 = 45 c = 3√5; c = 3√5/7