QUESTION 1
Given foci to be
[tex](\pm1,0)[/tex]
[tex]c = \pm1[/tex]
and
co-vertices
[tex](0, \pm2)[/tex]
[tex]b = \pm \: 2[/tex]
[tex] {b}^{2} = 4[/tex]
This implies the minor axis of the ellipse is on the y-axis and the major axis is on the x-axis.
We use the equation
[tex] {a}^{2} - {b}^{2} = {c}^{2} [/tex]
to determine the vertices.
[tex] {a}^{2} - {( \pm2)}^{2} = { ( \pm1)}^{2} [/tex]
[tex] {a}^{2} - 4 = 1[/tex]
[tex] {a}^{2} = 5[/tex]
The equation of the ellipse is given by;
[tex] \frac{ {x}^{2} }{ {a}^{2} } + \frac{ {y}^{2} }{ {b}^{2} } = 1[/tex]
[tex] \frac{ {x}^{2} }{ 5} + \frac{ {y}^{2} }{4} = 1[/tex]
QUESTION 2
The given ellipse has
Foci
[tex](0,\pm2)[/tex]
[tex]c = \pm2[/tex]
and co-vertices,
[tex](\pm1,0)[/tex]
[tex]b = \pm1[/tex]
[tex] {b}^{2} = 1[/tex]
We use the equation,
[tex] {a}^{2} - {b}^{2} = {c}^{2} [/tex]
to determine the vertices.
[tex]{a}^{2} - { (\pm1)}^{2} = {( \pm2)}^{2} [/tex]
[tex]{a}^{2} - 1= 4[/tex]
[tex]{a}^{2} =5[/tex]
The major axis of the ellipse is on the y-axis this time.
The equation is given by;
[tex] \frac{ {x}^{2} }{ {b}^{2} } + \frac{ {y}^{2} }{ {a}^{2} } = 1[/tex]
[tex]\frac{ {x}^{2} }{ 4} + \frac{ {y}^{2} }{ 5} = 1[/tex]
QUESTION 3
The given ellipse has
Foci
[tex](0,\pm4)[/tex]
and co-vertices,
[tex](\pm4,0)[/tex]
[tex]b = \pm4[/tex]
[tex] {b}^{2} = 16[/tex]
[tex]{a}^{2} - {b}^{2} = {c}^{2} [/tex]
[tex]{a}^{2} - 16= 16[/tex]
[tex]{a}^{2} = 32[/tex]
The major axis is on the y-axis.
The equation is,
[tex]\frac{ {x}^{2} }{ {b}^{2} } + \frac{ {y}^{2} }{ {a}^{2} } = 1[/tex]
[tex]\frac{ {x}^{2} }{16} + \frac{ {y}^{2} }{32 } = 1[/tex]