Answer
(a)
Center: $C(2,1)$
Vertices: $V_1(-1,1)$ and $V_2(5,1)$
Foci: $F_1(2-\sqrt {5},1)$ and $F_2(2+\sqrt {5},1)$
(b)
Length of the major axis:
$2a=6$
Length of the minor axis:
$2b=4$
(c)
Work Step by Step
Equation of an ellipse with center at $(h,k)$ (major axis is horizontal):
$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$
$\frac{(x-2)^2}{9}+\frac{(y-1)^2}{4}=1$
$\frac{(x-2)^2}{3^2}+\frac{(y-1)^2}{2^2}=1$
$h=2$
$k=1$
Center: $C(2,1)$
$a=3$
$b=2$
$c^2=a^2-b^2=9-4=5$
$c=\sqrt {5}$
The given equation can be obtained by shifting
$\frac{x^2}{3^2}+\frac{y^2}{2^2}=1$
right 2 units and upward 1 unit. In this equation:
Vertices: $V(±a,0)$
$V_1(3,0)$ and $V_2(-3,0)$
Foci: $F(±c,0)$
$F_1(\sqrt {5},0)$ and $F_2(-\sqrt {5},0)$
Now, shift these points 2 units to the right and 1 unit upward:
Vertices: $V_1(-1,1)$ and $V_2(5,1)$
Foci: $F_1(2-\sqrt {5},1)$ and $F_2(2+\sqrt {5},1)$
Length of the major axis:
$2a=6$
Length of the minor axis:
$2b=4$
