A hyperbola has vertices (4,0) and one focus (5,0). What is the standard-form equation of the hyperbola?
Since the vertices are on the x-axis, we know that this is a horizontal hyperbola. The distance from the center to each vertex is 4 units, so the center of the hyperbola is at (4, 0). The distance from the center to one focus is 5 units. We can use this information to find c, the distance from the center to the focus:
c² = a² + b²
5² = 4² + b²
25 = 16 + b²
b² = 9
b = 3
The standard form equation for a horizontal hyperbola with center at (h, k) and vertices a units to the left and right of the center is:
(x - h)² / a² - (y - k)² / b² = 1
Plugging in our values, we get:
(x - 4)² / 16 - y² / 9 = 1
So the standard-form equation of the hyperbola is:
(x - 4)² / 16 - y² / 9 = 1
To find the standard-form equation of a hyperbola with given vertices and one focus, we need to determine the values of the variables in the general equation of a hyperbola.
The standard-form equation of a hyperbola centered at the origin (0,0) is:
((x-h)^2)/a^2 - ((y-k)^2)/b^2 = 1
Where (h,k) is the center of the hyperbola, and 'a' and 'b' represent the distances from the center to the vertices and foci, respectively.
Given that the hyperbola has vertices (4,0) and one focus (5,0), we can determine the values for 'h', 'k', 'a', and 'b' as follows:
The distance from the center (h,k) to the vertices is 'a', so a = 4 - 0 = 4.
The distance from the center (h,k) to the focus is 'c', and this is equal to a + e, where 'e' is the eccentricity. Since the hyperbola is centered at (0,0), we can deduce that h = 0 and k = 0. Thus, the focus is (c, 0) = (5,0).
Therefore, c = 5 - 0 = 5, and a = 4.
To find 'b', we can use the relationship between 'a', 'b', and 'c' in a hyperbola equation, which is given by the equation:
c^2 = a^2 + b^2
Substituting the values, we have:
5^2 = 4^2 + b^2
25 = 16 + b^2
b^2 = 25 - 16
b^2 = 9
b = sqrt(9)
b = 3
Now that we have determined the values of 'a', 'b', and 'c', we can write the standard-form equation of the hyperbola as:
((x-0)^2)/4^2 - ((y-0)^2)/3^2 = 1
x^2/16 - y^2/9 = 1
Therefore, the standard-form equation of the hyperbola is x^2/16 - y^2/9 = 1.