Final answer:
The equation of the hyperbola with one asymptote given by y= 12x/5 +31 and a transverse axis of length 24 located on x=−3 can be found using the formula for a hyperbola with vertical transverse axis. The equation is (y - k)² / 12² - (x - (-3))² / 12² = 1, where (h, k) represents the coordinates of the center of the hyperbola, 'a' is the distance from the center to the vertices, and 'b' is the distance from the center to the foci.
Step-by-step explanation:
The equation of a hyperbola with one asymptote given by y= 12x/5 +31 and a transverse axis of length 24 located on x=−3 can be found by using the formula for a hyperbola with vertical transverse axis.
The standard equation for a hyperbola with vertical transverse axis is:
(y - k)² / a² - (x - h)² / b² = 1
In this equation, (h, k) represents the coordinates of the center of the hyperbola, 'a' represents the distance from the center to the vertices, and 'b' represents the distance from the center to the foci.
Since the transverse axis has a length of 24, we know that 'a' is equal to half of the length, which is 12. Additionally, since the transverse axis is located on x = -3, we know that the center of the hyperbola is (-3, k).
We can now substitute these values into the equation:
(y - k)² / 12² - (x - (-3))² / b² = 1
Now, we need to find the value of 'b'. Since one of the asymptotes is given by y = 12x/5 + 31, we can find the slope of this asymptote, which will be equal to (b / a) or (distance from asymptote to center / distance from center to foci). In this case, the slope of the asymptote is 12/5.
Therefore, we have the equation 12/5 = b / 12. Solving for 'b', we get b = 60/5 = 12.
Now we can substitute the values of 'a', 'b', and the coordinates of the center into the equation to get the final equation of the hyperbola:
(y - k)² / 12² - (x - (-3))² / 12² = 1