Standard form of hyperbola: Hyperbola: Vertices: (9,9), (9, -7) Foci: (9, 1 + 4 sqrt 13), (9, 1 - 4 sqrt 13)?
Standard form Ellipse: vertices: (2 sqrt 10, -10), (-2 sqrt 10, -10) foci: (sqrt 30, -10), (-sqrt 30, -10)?
the center is at (9,1)
a=8
c=4?13
b^2 = c^2-a^2 = 208-64=144
axis is vertical,so
(y-1)^2/144 - (x-9)^2/64 = 1
http://www.wolframalpha.com/input/?i=hyperbola+(y-1)%5E2%2F144+-+(x-9)%5E2%2F64+%3D+1
do the other in like wise.
Oops I got a and b reversed.
http://www.wolframalpha.com/input/?i=hyperbola+(y-1)%5E2%2F64+-+(x-9)%5E2%2F144+%3D+1
To find the standard form of a hyperbola or an ellipse, you need to have information about the vertices and the foci. Let's start by looking at the hyperbola.
For a hyperbola, the standard form equation is:
((x-h)^2)/a^2 - ((y-k)^2)/b^2 = 1 or ((y-k)^2)/b^2 - ((x-h)^2)/a^2 = 1
where (h,k) is the center of the hyperbola and 'a' is the distance from the center to the vertices, while 'b' is the distance from the center to the foci.
From the given information, we can extract the following:
Vertices: (9,9), (9, -7)
Foci: (9, 1 + 4 √13), (9, 1 - 4 √13)
Comparing the given information with the standard form equation, we can deduce the following:
Center: (h,k) = (9, 1)
Distance from the center to the vertices: a = 8 (distance from (9,9) to (9,-7))
Distance from the center to the foci: c = 4√13
Now, we need to find the value of 'b'. We can use the relationship between 'a', 'b', and 'c' in a hyperbola:
c^2 = a^2 + b^2
Substituting the known values:
(4√13)^2 = 8^2 + b^2
52 = 64 + b^2
b^2 = -12
Since 'b' squared cannot be negative, we conclude that there is an error in the information provided for the hyperbola.
Now, let's move to the ellipse.
For an ellipse, the standard form equation is:
((x-h)^2)/a^2 + ((y-k)^2)/b^2 = 1
Using the given information:
Vertices: (2√10, -10), (-2√10, -10)
Foci: (√30, -10), (-√30, -10)
Comparing with the standard form equation, we can deduce the following:
Center: (h,k) = (0, -10)
Distance from the center to the vertices: a = 2√10
Distance from the center to the foci: c = √30
Again, we will find the value of 'b' using the relationship between 'a', 'b', and 'c':
c^2 = a^2 - b^2
Substituting the known values:
(√30)^2 = (2√10)^2 - b^2
30 = 40 - b^2
b^2 = 10
Therefore, the standard form equation for the given ellipse is:
((x-0)^2)/(2√10)^2 + ((y-(-10))^2)/10 = 1
or
x^2/(4√10) + (y+10)^2/10 = 1