a hyperbola has vertices (+-5,0) and one focus (6,0) what is the standard form equation of the hyperbola.
The standard form equation of the hyperbola:
x² / a² - y² / b² = 1
a = 5,
a² = 25
c = 6
c² = a² + b²
36 = 25 + b²
b² = 36 - 25
b² = 11
Answer:
x²/25 - y²/11 = 1
(x^2)/25 - (y^2)/11 = 1
Why did the hyperbola go see a psychiatrist? Because it had an Identity Complex!
But in all seriousness, let's help you out. To find the equation of a hyperbola given its vertices and focus, we can start by determining some key information.
The distance between the center and vertices is given by "a" (in this case, a = 5), and the distance between the center and focus is given by "c" (in this case, c = 6).
To find "b," we can use the relationship between "a," "b," and "c" in a hyperbola: c^2 = a^2 + b^2.
Plugging in the values, we get 6^2 = 5^2 + b^2, which simplifies to 36 = 25 + b^2. By subtracting 25 from both sides, we find b^2 = 11.
Now we have all the necessary information to form the standard form equation of a hyperbola with center (h, k), where h and k are the coordinates of the center:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1.
In this case, the center of the hyperbola is (0, 0) because the y-coordinate remains constant.
Substituting the values we determined, the equation becomes:
(x - 0)^2 / 5^2 - (y - 0)^2 / (√11)^2 = 1.
Simplifying further, we get:
x^2 / 25 - y^2 / 11 = 1.
So, the standard form equation of the hyperbola is x^2 / 25 - y^2 / 11 = 1.
To determine the standard form equation of a hyperbola, you need the coordinates of the vertices and the coordinates of one focus.
Given:
Vertices: (+-5,0)
Focus: (6,0)
1. Start by determining the center of the hyperbola which is the midpoint between the vertices. The x-coordinate of the center is the average of the x-coordinates of the vertices, and the y-coordinate remains the same. In this case, the center is (0, 0).
2. Determine the distance between the center and one of the vertices. In this case, the distance is 5 units.
3. Since the vertices lie on the x-axis, the equation of the hyperbola takes the form:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
where (h, k) is the center, and a and b are the distances between the center and the vertices and between the center and the foci, respectively.
4. Substitute the given values into the equation.
Center: (0,0)
a = 5
(x - 0)^2 / 5^2 - (y - 0)^2 / b^2 = 1
(x^2 / 25) - (y^2 / b^2) = 1
5. To find the value of b, we can use the relationship between a, b, and the distance between the center and the focus, c, which is given by:
c^2 = a^2 + b^2
Substitute the known values:
(6 - 0)^2 = 5^2 + b^2
36 = 25 + b^2
b^2 = 36 - 25
b^2 = 11
6. Substitute the value of b^2 back into the equation:
(x^2 / 25) - (y^2 / 11) = 1
Therefore, the standard form equation of the hyperbola is:
(x^2 / 25) - (y^2 / 11) = 1
To find the standard form equation of a hyperbola with the given information, we can start with the general form equation of a hyperbola:
(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)
where (h, k) represents the center of the hyperbola.
In this case, we can observe that the center of the hyperbola is at (0, 0) since the vertices are symmetrically placed about the origin.
The distance between the center and each vertex is given by 'a'. In this case, a = 5 since the vertices are at (+-5, 0).
The distance between the center and the focus is given by 'c'. In this case, c = 6 since the focus is at (6, 0).
Using these values, we can now determine the value of 'b' using the relationship between 'a', 'b', and 'c' in a hyperbola:
c^2 = a^2 + b^2
Solving for 'b', we have:
b^2 = c^2 - a^2
b^2 = 6^2 - 5^2
b^2 = 36 - 25
b^2 = 11
Now we can plug these values into the general form equation, based on the orientation of the hyperbola:
For a horizontal hyperbola:
(x - 0)^2 / 5^2 - (y - 0)^2 / 11 = 1
Simplifying, we get:
x^2/25 - y^2/11 = 1
Thus, the standard form equation of the hyperbola is x^2/25 - y^2/11 = 1.