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+2)²/36 + (y-5)²/36 = 1
2. (x+7)²/27 + (y+2)²/20 = 1

Answer 1

1 is a circle with radius 6 and center (-2,5)

general equation for a circle is (x-h)^2 + (y-k)^2 = r^2 where (h,k) is the center and r= radius. diameter = 2r = 2(6) = 12

Multiply the given equation by 36 to get (x+2)^2 + (y-5)^2 = 36 = 6^2, then (h,k) =(-2,5) and r= sqr36 = 6

2 is an ellipse

length of major axis: 2sqr27 = 6sqr3

length of minor axis is 2sqr20 = 4sqr5

direction of the major axis is horizontal

eccentricity = e = c/a where c=sqr(a^2-b^2) = sqr(27-20) = sqr7

e=sqr7/sqr27 =(sqr(7/3))/3

or e=sqr(1-b^2/a^2) = sqr(1-20/27) = sqr(7/27) = (sqr(7/3))/3=about 0.509

center of the ellipse is (-7,-2)

foci are (h+c,k) and (h-c,k) = (-7+sqr7, -2) and (-7-sqr7, -2)