Can someone please explain to me how to get the 16 in this equation. I get how to get everything else but i don't know how to get the 16.
A hyperbola with foci: (0,5) (0,-5)and vertices: (0,3) (0,-3). So i got y^2/9-x^2/16=1. but how do you get the 16?
center is at (0,0) so the form is:
y^2/a^2 -x^2/b^2 = 1
distance center to focus = 5 = sqrt(a^2+b^2)
distance center to vertex = 3 = a
so
5^2 = (3^2 + b^2)
25 = 9 + b^2
b^2 = 16
thanks man
To understand how to get the value of 16 in the equation, let's first review some key properties of a hyperbola:
1. The standard form of a hyperbola centered at the origin is given by:
(y^2 / a^2) - (x^2 / b^2) = 1
where 'a' represents the distance from the center to the vertices along the y-axis, and 'b' represents the distance from the center to the vertices along the x-axis.
2. The distance between the center and each focus along the y-axis is represented by 'c', which is related to 'a' and 'b' by the equation:
c^2 = a^2 + b^2
Now let's apply these properties to solve for the value of 16 in the equation:
1. From the given information, we know that the hyperbola has foci at (0,5) and (0,-5). This means the distance 'c' is 5.
2. The vertices are located at (0,3) and (0,-3). This tells us that the distance 'a' is 3.
3. Using the formula c^2 = a^2 + b^2, we can substitute the known values:
5^2 = 3^2 + b^2
25 = 9 + b^2
4. Rearranging the equation to isolate 'b^2', we have:
b^2 = 25 - 9
b^2 = 16
Now we can substitute the values of 'a' and 'b^2' into the standard form of the hyperbola equation:
(y^2 / 9) - (x^2 / 16) = 1
Hence, the value of 16 in the equation is obtained by calculating the difference between the squares of 'c' and 'a' in the formula c^2 = a^2 + b^2.