HYPERBOLA Q!! Find an equation for a horizontal hyperbola with vertices that are 20 units apart from each other and foci 30 units apart each other?? im lost on this any help is appreciated !!!
thankyou!!!! i think you meant 225-100 = 125, but other than that this helped me out sm!!! thanks!!!
see, you understand this, or else you would not have caught my typo.
Place the vertices at (10,0) and (-10,0) and the foci
at (15,0) and (-15,0).
That will place the centre conveniently at (0,0)
and a = 10, c = 15
for a hyperbola:
a^2 + b^2 = c^2
100 + c^2 = 225
b^2 = 115
equation:
x^2/100 - y^2/115 = 1
Now, wasn't that easy?
To find the equation of a hyperbola with given vertices and foci, you can use the standard form equations for horizontal hyperbolas. The standard form equation for a horizontal hyperbola is:
((x - h)^2 / a^2) - ((y - k)^2 / b^2) = 1
where (h, k) is the center of the hyperbola, and a and b are the lengths of the semi-major and semi-minor axes, respectively.
In this case, the vertices are 20 units apart, so a = 20/2 = 10. The foci are 30 units apart, so c = 30/2 = 15.
The relationship between a, b, and c in a hyperbola is given by the equation c^2 = a^2 + b^2. Therefore, we can substitute the values of c and a into this equation to find b:
b^2 = c^2 - a^2
b^2 = 15^2 - 10^2
b^2 = 225 - 100
b^2 = 125
Now that we have the values of a and b, we can determine the center of the hyperbola, which is halfway between the vertices. In this case, the center is at (h, k) = (0, 0).
Finally, substitute the values of h, k, a, and b into the standard equation of a horizontal hyperbola to get the final equation:
((x - 0)^2 / 10^2) - ((y - 0)^2 / √125)^2 = 1
Simplifying this equation gives the equation of the hyperbola:
x^2 / 100 - y^2 / 125 = 1