What is the equation of hyperbola whose foci at f(-8,0) & length of transverse axis is 6?
To find the equation of a hyperbola given the foci and the length of the transverse axis, we can follow these steps:
Step 1: The coordinates of the foci are given as f(-8, 0).
Step 2: The distance between the foci and the center of the hyperbola is equal to the value of "c" (which represents the distance from the center to each focus).
Step 3: The length of the transverse axis is given as 6. The transverse axis is the line segment passing through the center of the hyperbola and having endpoints on the curve. It is twice the distance of "a" (which represents the distance from the center to each vertex).
Let's label the center of the hyperbola as (h, k). We know that the coordinates of each focus are (h - c, k) and (h + c, k).
From the given foci, we can calculate "c" as follows:
c = 8
Since the length of the transverse axis is given as 6, we know that 2a = 6, which means a = 3.
Step 4: Now we have all the necessary values to write the equation of the hyperbola.
The equation of a hyperbola is given by:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
In our case, h = 0, k = 0, a = 3, and we need to find b.
To find b, we can use the relationship between a, b, and c in a hyperbola: a^2 + b^2 = c^2.
Plugging in the values:
3^2 + b^2 = 8^2
9 + b^2 = 64
b^2 = 64 - 9
b^2 = 55
b = sqrt(55)
Thus, the equation of the hyperbola is:
x^2 / 3^2 - y^2 / (sqrt(55))^2 = 1
Simplifying further:
x^2 / 9 - y^2 / 55 = 1
Therefore, the equation of the hyperbola with foci at f(-8,0) and a transverse axis length of 6 is:
x^2 / 9 - y^2 / 55 = 1.
To find the equation of a hyperbola given the foci and the length of the transverse axis, you can use the formula:
c = sqrt(a^2 + b^2)
where c is the distance between the center of the hyperbola and each focus, and a is the distance from the center to a vertex.
In this case, you are given that the foci are located at (-8, 0). Since the x-coordinate of each focus is negative, the center of the hyperbola must be to the right of the origin.
The distance between the center and each focus is c. Substituting the coordinates of one of the foci into the equation, we get:
c = sqrt((-8 - h)^2 + (0 - k)^2)
where (h, k) represents the coordinates of the center.
Since the foci are (-8, 0), we have:
c = sqrt((-8 - h)^2 + (0 - k)^2)
The length of the transverse axis is 6, which is equal to 2a. Therefore, a = 6/2 = 3.
Now, we can substitute the values into the equation for c:
c = sqrt(((-8) - h)^2 + (0 - k)^2)
Since the distance between the center and each focus is c, and the distance between the center and each vertex is a, we have:
c = a
Therefore, we can write:
sqrt(((-8) - h)^2 + (0 - k)^2) = 3
Squaring both sides of the equation, we get:
((-8) - h)^2 + (0 - k)^2 = 3^2
Simplifying further, we get:
(h + 8)^2 + k^2 = 9
So, the equation of the hyperbola is:
(h + 8)^2 + k^2 = 9