Final answer:
To find the equation of the hyperbola, we can use the equation y = ax + bx² and the given information. By substituting the coordinates of the point (5,2) and the distance between the vertices, we can solve for the values of a and b. These values can then be used to write the equation of the hyperbola.
Step-by-step explanation:
The equation of a hyperbola of the form y = ax + bx² can be used to find the equation of the hyperbola in this question. The length of the transverse axis of the hyperbola is given as 7, which means the distance between the two vertices is 7. Since the hyperbola passes through the point (5,2), we can use this information to find the values of a and b.
By substituting the coordinates of the point (5,2) into the equation y = ax + bx², we get 2 = 5a + 25b. This gives us one equation. Additionally, since the distance between the vertices is 7, we know that the difference between the y-coordinates of the vertices is also 7. Substituting the coordinates of the vertices into the equation, we get 7 = 2a + 2.25b. This gives us a second equation.
Solving these two equations simultaneously will give us the values of a and b, which we can then use to find the equation of the hyperbola.