Find the equation of the hyperbola satisfying the given conditions.
Vertices at (5, 1) and (-7, 1); asymptotes (y - 1) = 1/3(x + 1) and (y - 1) = -1/3(x + 1)
centre is the midpoint of the vertices:
C(-1,1)
our a value is 6, and we don't know b
but we know b/a = 1/3
b/6 = 1/3
3b = 6
b = 2
(x+1)^2 / 36 - (y-1)^2/4 = 1
or
(x+1)^2 - 9(y-1)^2 = 36
verification:
http://www.wolframalpha.com/input/?i=plot+(x%2B1)%5E2+-+9(y-1)%5E2+%3D+36
what is the standard equation given asymptotes are y=1/3(x+5) and y=-1/3(x-7) with foci (1,12)
To find the equation of the hyperbola, we need to use the given information. The standard form of the equation of a hyperbola with center (h, k) is:
((x - h)^2 / a^2) - ((y - k)^2 / b^2) = 1 for a horizontal hyperbola, or
((y - k)^2 / a^2) - ((x - h)^2 / b^2) = 1 for a vertical hyperbola.
Let's start by finding the center of the hyperbola. The center is the midpoint between the vertices, which are given as (5, 1) and (-7, 1).
Using the midpoint formula, the x-coordinate of the center (h) can be found by adding the x-coordinates of the vertices and dividing by 2:
h = (5 + (-7)) / 2 = -2 / 2 = -1.
Since the y-coordinate of the vertices is the same (1), the y-coordinate of the center (k) is also 1.
So, the center of the hyperbola is (-1, 1).
Next, let's find the values of a and b, which represent half the distance between the center and the vertices along the x and y directions, respectively.
The distance between the center and one of the vertices along the x-axis (a) is the absolute value of the difference between the x-coordinates of the vertices:
a = |5 - (-7)| / 2 = 12 / 2 = 6.
Similarly, the distance between the center and one of the vertices along the y-axis (b) is the absolute value of the difference between the y-coordinates of the vertices, which is 0 since they have the same y-coordinate.
Since the asymptotes are given, we can determine the value of a from them. The slopes of the asymptotes are given as 1/3 and -1/3. The value of a is the reciprocal of the slope.
So, a = 1 / (1/3) = 3.
Now that we have the center (h, k) = (-1, 1) and the values of a = 6, we can write the equation of the hyperbola in standard form for a horizontal hyperbola:
((x - h)^2 / a^2) - ((y - k)^2 / b^2) = 1.
Substituting the values, we get:
((x - (-1))^2 / 6^2) - ((y - 1)^2 / b^2) = 1.
Simplifying it further, we have:
(x + 1)^2 / 36 - (y - 1)^2 / b^2 = 1.
Since the asymptotes represent the lines that the hyperbola approaches as x goes to infinity, they help us determine the value of b^2. Let's use the equation of one of the asymptotes to do that.
The equation given is (y - 1) = 1/3(x + 1).
Expanding it, we have:
3y - 3 = x + 1.
Rearranging it, we get:
x = 3y - 4.
Comparing this equation with the standard form of a hyperbola, we can see that b^2 is the coefficient of y^2 in the equation.
So, b^2 = 1.
Finally, we can write the equation of the hyperbola satisfying the given conditions as:
((x + 1)^2 / 36) - ((y - 1)^2 / 1) = 1.