Question

can someone please help me with these questions?? I dont get them at all! i would appreciate the help.

1.) find the standard form or the equation of the ellipse with vertices (±9,0) and eccentricity e=1/3.

2.) Find the standard form of the equation of the hyperbola with the given characteristics.
vertices: (2, - 7), (2, - 1),
foci: (2, - 9), (2, 1)

3.) Find the standard form of the equation of the ellipse with the given characteristics.
center: (-1, 7)
a = 3c
foci: (- 2, 7), (0, 7)

Answer 1

Let 2a be major axis
Because vertices are (-9,0) and (9,0)
hence centre is (0,0) and 2a=18
a=9
\[e=\frac{ 1 }{ 3 }\]
Let b be the semi minor axis.\[b ^{2}=a ^{2}\left( 1-e ^{2} \right),b ^{2}=9^{2}\left( 1-\left( \frac{ 1 }{ 3 } \right)^{2} \right)\]
\[b ^{2}=81\left( 1-\frac{ 1 }{9 } \right)=81*\frac{ 8 }{ 9 }=72=36*2\]
\[b=6\sqrt{2}\]
eq. of ellipse is
\[\frac{ x ^{2} }{ 9^{2} }+\frac{ y ^{2} }{ \left( 6\sqrt{2} \right)^{2} }=1\]

Answer 2

3.

\[distance \between two foci 2c=\sqrt{\left( 0-\left( -2 \right) \right)^{2}+\left( 7-7 \right)^2}\]=2
c=1
a=3c=3*1=3
\[b ^{2}=a ^{2}-c ^{2}=3^{2}-1^{2}=8,b=2\sqrt{2}\]
centre is (-1,7)
eq. of ellipse is
\[\frac{ \left( x-\left( -1 \right) \right)^{2} }{ 3^{2} }+\frac{ \left( y-7 \right)^{2} }{ \left( 2\sqrt{2} \right)^{2} }=1 \]
\[\frac{ \left( x+1 \right)^{2} }{ 3^{2} }+\frac{ \left( y-7 \right)^{2} }{ \left( 2\sqrt{2} \right)^{2} }=1\]

Answer 3

2.

vertices (2,-7),(2,-1)
foci (2,1),(2,-9)
centre is mid point of (2,-1),(2,-7)
\[=\left( \frac{ 2+2 }{ 2 },\frac{ -1+(-7) }{ 2 } \right)=\left( 2,-4 \right)\]
semi major axis a=distance between centre and one vertex
\[a=\sqrt{\left( 2-2 \right)^{2}+\left( -4-(-7) \right)^{2}}=3\]
c=distance between centre and one focus
\[c=\sqrt{\left( 2-2 \right)^{2}+\left( -4-(-9) \right)^{2}}=5\]
\[c ^{2}=a ^{2}+b ^{2},5^{2}=3^{2}+b ^{2},b ^{2}=25-9=16,b=4\]
eq.of hyperbola is
\[\frac{ \left( y-k \right)^{2} }{ a ^{2} }-\frac{ \left( x-h \right)^{2} }{b ^{2} }=1\]
\[-\frac{ \left( x-2 \right)^{2} }{ 4^{2} }+\frac{ \left( y+4 \right)^{2} }{3^{2} }=1\]