Final answer:
To find the standard form of the hyperbola, we use the vertices to determine the center and the transverse axis length, then use the additional point (0,5) to find the missing value and complete the equation.
Step-by-step explanation:
The question involves finding the standard form of the equation of a hyperbola with given vertices and a point through which it passes. The vertices are (2,3) and (2,-3), which are vertically aligned, indicating that the transverse axis is vertical. To find the standard form of the equation, we use the formula:
\frac{{(y-k)^2}}{{a^2}} - \frac{{(x-h)^2}}{{b^2}} = 1
Since the vertices are vertical, k will be the y-coordinate of the center of the hyperbola, a is the distance from the center to a vertex along the y-axis, and h will be the x-coordinate of the center. The distance between the vertices is 6, so a is 3. The center of the hyperbola is at (2, 0). Therefore, our equation starts to take shape as:
\frac{{(y-0)^2}}{{3^2}} - \frac{{(x-2)^2}}{{b^2}} = 1
We still need to find b by using the additional point (0,5) that lies on the hyperbola. Substituting this point into the equation, we solve for b^2. Ultimately, we get the standard form of the hyperbola's equation.