QUESTION 1
The given ellipse has equation:
[tex]4 {x}^{2} + 25 {y}^{2} = 100[/tex]
Divide through by 100 to get;
[tex] \frac{ {x}^{2} }{25} + \frac{ {y}^{2} }{4} = 1[/tex]
The ellipse has it major axis on the x-axis.
[tex] {a}^{2} = 25[/tex]
[tex]a = \pm5[/tex]
The vertices is
(-5,0) and (5,0).
Also
[tex] {b}^{2} = 4[/tex]
[tex]b = \pm2[/tex]
The co-vertices are,
(0,-2), (0,2).
We use
[tex] {a}^{2} - {b}^{2} = {c}^{2} [/tex]
[tex] {c}^{2} = {( \pm5)}^{2} - {( \pm2)}^{2} [/tex]
[tex] {c}^{2} = 25 - 4[/tex]
[tex] {c}^{2} =21[/tex]
[tex]c = \pm \sqrt{21} [/tex]
The foci:
[tex](\pm \sqrt{21} ,0)[/tex]
We plot all these points and graph our ellipse.
See attachment
QUESTION 2.
If the ellipse has a vertex at
(0,-8) and a focus at (0,4) then
[tex] {a}^{2} = 64[/tex]
and
[tex] {c}^{2} = 16[/tex]
we can use
[tex]{a}^{2} - {b}^{2} = {c}^{2} [/tex]
to determine the value if b.
[tex] {( - 8)}^{2} - {b}^{2} = {4}^{2} [/tex]
[tex] 64- {b}^{2} = 16[/tex]
[tex]{b}^{2} = 64 - 16[/tex]
[tex] {b}^{2} = 48[/tex]
The equation of the ellipse is
[tex] \frac{ {x}^{2} }{ {b}^{2} } + \frac{ {y}^{2} }{ {a}^{2} } = 1[/tex]
[tex] \frac{ {x}^{2} }{48} + \frac{ {y}^{2} }{64} = 1[/tex]
