I really need help
Find the equation of the hyperbola
1.Vertices(-4,0)(4,0)
Asymptotes: y=1/3x, y=-1/3x
2.Vertices(-6,0)(6,0)
Asymptotes: y=4/3x, y=-4/3x
For the first one i got ((x-9)^2+y^2-(x+9)^2+y^2+16=0
good heavens, whatever did you do to get that equation ?
I made a sketch, clearly the centre is at (0,0)
My sketch contains the containing rectangle formed by the asymptotes and the vertices
so a = 4, and from the asymptotes, b/a = 1/3
b/4 = 1/3
b = 4/3
equation is:
x^2 /16 - y^2 /(16/9) = 1
or
x^2 - 9y^2 = 16
do the same for the 2nd question
To find the equation of a hyperbola, you need to know the coordinates of the vertices and the equations of the asymptotes.
For the first example, we have the vertices at (-4,0) and (4,0) and the asymptotes y = (1/3)x and y = -(1/3)x.
Step 1: Determine the center of the hyperbola
The center of the hyperbola is the midpoint between the vertices. In this case, the midpoint is (0,0), so the center is at (h, k) = (0,0).
Step 2: Determine the value of 'a'
The distance from the center to either vertex is 'a'. In this case, the distance from the center to either vertex is 4 units, so 'a' is equal to 4.
Step 3: Determine the value of 'b'
The distance from the center to either of the intersections between the asymptote lines and the hyperbola is 'b'. In this case, the distance from the center to the intersection point is 3 units. Since the asymptotes have a slope of 1/3, the value of 'b' is (1/3) * 3 = 1.
Step 4: Determine the equation of the hyperbola
The general equation for a hyperbola centered at (h, k) can be written as:
((x - h)^2) / (a^2) - ((y - k)^2) / (b^2) = 1
In this case, the equation becomes:
((x - 0)^2) / (4^2) - ((y - 0)^2) / (1^2) = 1
Simplifying, we get:
x^2 / 16 - y^2 / 1 = 1
So, the equation of the hyperbola is:
x^2 / 16 - y^2 = 1