Answer
(a)
Center: $C(0,1)$
Vertices: $V_1(-5,1)$ and $V_2(5,1)$
Foci: $F_1(-\sqrt {21},1)$ and $F_2(\sqrt {21},1)$
(b)
Length of the major axis:
$2a=10$
Length of the minor axis:
$2b=4$
Work Step by Step
Equation of an ellipse with center at $(h,k)$ (major axis is horizontal):
$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$
$4x^2+25y^2-50y=75$
$4x^2+25y^2-50y+25=75+25$
$4x^2+25(y-1)^2=100$
$\frac{x^2}{25}+\frac{(y-1)^2}{4}=1$
$\frac{(x-0)^2}{5^2}+\frac{(y-1)^2}{2^2}=1$
$h=0$
$k=1$
Center: $C(0,1)$
$a=5$
$b=2$
$c^2=a^2-b^2=25-4=21$
$c=\sqrt {21}$
The given equation can be obtained by shifting
$\frac{x^2}{5^2}+\frac{y^2}{2^2}=1$
upward 1 unit. In this equation:
Vertices: $V(±a,0)$
$V_1(-5,0)$ and $V_2(5,0)$
Foci: $F(±c,0)$
$F_1(-2\sqrt {3},0)$ and $F_2(2\sqrt {3},0)$
Now, shift these points 1 unit upward:
Vertices: $V_1(-5,1)$ and $V_2(5,1)$
Foci: $F_1(-\sqrt {21},1)$ and $F_2(\sqrt {21},1)$
Length of the major axis:
$2a=10$
Length of the minor axis:
$2b=4$
