Find the equation of the hyperbola.
Transverse axis parallel to the x-axis, center at (5,1), the rectangle on the axes of the hyperbola of area 48 and distance betwern foci 10.
i do nt know how . sorry . pls do help me ..
To find the equation of the hyperbola, we need to use the given information about the hyperbola's center, transverse axis, area of the rectangle, and distance between the foci.
1. Determine the values of a and b:
Since the transverse axis is parallel to the x-axis, the equation of the hyperbola takes the form:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
Given that the center of the hyperbola is at (5, 1), the values of h and k are 5 and 1, respectively.
2. Calculate the value of a:
The area of the rectangle formed by the hyperbola's asymptotes is equal to 2ab.
Since we have the area (48), we can find a relationship between a and b:
2ab = 48
3. Find the distance between the foci (c):
The distance between the foci is given as 10.
We can now use the relationship between a, b, and c to find a.
a^2 + b^2 = c^2
a^2 + b^2 = 10^2
4. Solve the equations to find the values of a and b:
We have two equations: 2ab = 48 and a^2 + b^2 = 100.
Using substitution, we can rewrite the second equation as b = sqrt(100 - a^2).
Plugging this into the first equation, we get: 2a(sqrt(100 - a^2)) = 48.
Simplify this equation and solve for a.
5. Once you have determined the value of a, substitute it back into the equation b = sqrt(100 - a^2) to find the value of b.
6. With the values of a and b, we can finally write the equation of the hyperbola in the form:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1, with (h, k) as the center and a and b as the distances from the center to the vertices along the x-axis and y-axis, respectively.
By following these steps, you should be able to find the equation of the hyperbola satisfying the given conditions.
To find the equation of the hyperbola, we need to gather the necessary information from the given problem.
1. The transverse axis is parallel to the x-axis.
2. The center is located at (5, 1).
3. The rectangle on the axes of the hyperbola has an area of 48.
4. The distance between the foci is 10.
First, let's find the length of the transverse axis (2a) using the formula:
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The area of the rectangle is given as 48, so if we assume the length (l) is the longer side of the rectangle and the width (w) is the shorter side, we have:
48 = l * w
Since we know that the rectangle is on the axes of the hyperbola, we can assume that the length is twice the distance from the center to the foci and the width is twice the distance from the center to the asymptotes.
Thus, l = 2 * (distance from center to foci) = 2 * 10 = 20.
Using the formula for the area, we find the width:
48 = 20 * w
Solving for w, we get w = 48/20 = 2.4.
Now, we can find the value of 'a' which represents half the length of the transverse axis:
a = l / 2 = 20 / 2 = 10.
Next, we need to find the value of 'c', which represents the distance from the center to each focus.
Since the distance between the foci is given as 10, we have:
2c = 10
c = 10 / 2 = 5.
The equation for the hyperbola with a transverse axis parallel to the x-axis, center at (h, k), is given by:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1,
where (h, k) is the center of the hyperbola.
Substituting the given values, we have:
(x - 5)^2 / 10^2 - (y - 1)^2 / b^2 = 1.
Now, we need to find the value of 'b'.
Using the relationship between 'a', 'b', and 'c' in a hyperbola:
c^2 = a^2 + b^2,
we can solve for 'b'.
(5)^2 = (10)^2 + b^2
25 = 100 + b^2
b^2 = 25 - 100 = -75.
Since 'b' is imaginary, there is no valid solution for this problem. Hence, the given conditions do not produce a valid hyperbola equation.