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 a hyperbola given its asymptotes and foci, we need to determine the center of the hyperbola, the distance between the center and the foci, and the value of 'a' (distance between the center and the vertex). Here are the step-by-step instructions:
Step 1: Determine the center of the hyperbola:
- The center of the hyperbola is the intersection point of the asymptotes.
- Set the equations of the asymptotes equal to each other:
1/3(x+5) = -1/3(x-7)
- Solve the equation for x:
x/3 + 5/3 = -x/3 + 7/3
x + 5 = -x + 7
2x = 2
x = 1
- Substitute the value of x into either asymptote equation to find the corresponding y-coordinate:
y = 1/3(1 + 5) = 2
- Thus, the center of the hyperbola is (1, 2).
Step 2: Find the distance between the center and the foci:
- Recall that the distance between the center and the foci is given by the equation c^2 = a^2 + b^2, where a is the distance between the center and the vertex, and b is the distance between the vertices and the foci.
- The center of the hyperbola is (1, 2), and one of the foci is (1, 12).
- Calculate the distance between the center and the foci using the distance formula:
c^2 = (x2 - x1)^2 + (y2 - y1)^2
c^2 = (1 - 1)^2 + (12 - 2)^2
c^2 = 0 + 10^2
c^2 = 100
c = 10
Step 3: Find the value of 'a':
- The value of 'a' is the distance between the center and the vertex.
- Since the asymptotes have slopes of ±1/3, the value of 'a' is equal to the distance between the center and any point on an asymptote.
- Choose a point on one of the asymptotes (for example, (0, -5)) and calculate the distance between the center (1, 2) and that point using the distance formula:
a = √[(x2 - x1)^2 + (y2 - y1)^2]
a = √[(0 - 1)^2 + (-5 - 2)^2]
a = √[(-1)^2 + (-7)^2]
a = √[1 + 49]
a = √50
a ≈ 7.07
Step 4: Determine the equation of the hyperbola:
- The standard equation of a hyperbola with center (h, k), asymptotes y = mx + b, and distance between the center and a focus c is given by:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
- Substituting the known values, we have:
(x - 1)^2 / (7.07)^2 - (y - 2)^2 / b^2 = 1
Please note that we still need to determine the value of 'b' in order to have a complete equation. This information is not provided in the question. If you have the value of 'b', you can substitute it into the equation. Otherwise, you might need more information to complete the equation.
To find the standard equation of a hyperbola given its asymptotes and foci, we can follow these steps:
Step 1: Determine the center of the hyperbola.
The center of the hyperbola is the midpoint between the foci. Using the given foci (1,12), we can find the center as follows:
Center = ((x-coordinate of focus1 + x-coordinate of focus2) / 2, (y-coordinate of focus1 + y-coordinate of focus2) / 2)
Center = ((1 + 7) / 2, (12 + 12) / 2)
Center = (4, 12)
Step 2: Determine the equation of the asymptotes.
The equation of the asymptotes for a hyperbola with center (h, k) and slopes given by m is y = mx + (k - mh) + c, where c is an arbitrary constant.
Given the asymptotes y = 1/3(x + 5) and y = -1/3(x - 7), we can determine the slopes:
For the first asymptote, the slope is m1 = 1/3.
For the second asymptote, the slope is m2 = -1/3.
Since the slopes of the asymptotes are reciprocals of each other, the hyperbola is symmetrical and the equation of the asymptotes can be written as follows:
y - k = ±(x - h) * √(a^2 + 1),
where (h, k) is the center, and a is the distance from the center to the vertex.
Plugging in the values, we have:
y - 12 = ±(x - 4) * √(a^2 + 1)
Step 3: Determine the value of "a".
In a hyperbola, the distance between the center and each vertex is given by "a". To find "a," we can use the distance formula between the center and the given foci:
Distance between center and focus = √((x2 - x1)^2 + (y2 - y1)^2)
Distance between center (4, 12) and focus (1, 12) = √((1 - 4)^2 + (12 - 12)^2)
Distance between center and focus = √((-3)^2 + 0^2)
Distance between center and focus = √(9 + 0)
Distance between center and focus = √9
Distance between center and focus = 3
Therefore, the value of "a" is 3.
Step 4: Write the equation of the hyperbola.
Using the value of "a" obtained in the previous step and substituting it into the equation of the asymptotes, we have:
y - 12 = ±(x - 4) * √(a^2 + 1)
y - 12 = ±(x - 4) * √(3^2 + 1)
y - 12 = ±(x - 4) * √(9 + 1)
y - 12 = ±(x - 4) * √(10)
y - 12 = ±√10 * (x - 4)
This is the equation of the hyperbola.
The asymptotes intersect at the center: (1,2)
The foci are at x=1, so the axis is vertical, giving
(y-2)^2/a^2 - (x-1)^2/b^2 = 1
If a focus is at (1,12), c=10
c^2 = a^2+b^2
The slope of the asymptotes is a/b = 1/3, so b=3a
c^2 = a^2+b^2 = 100, so
a^2=10, b^2=90
(y-2)^2/10 - (x-1)^2/90 = 1
verify at
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