Answer:
[tex]\dfrac{y^2}{36}-\dfrac{x^2}{45}=1.[/tex]
Step-by-step explanation:
Since vertices and foci lie on the y-axis, the equation of the hyperbola is
[tex]\dfrac{y^2}{b^2}-\dfrac{x^2}{a^2}=1.[/tex]
If the vertices are at points (0,±6), then [tex]b=6.[/tex]
If the foci are at points (0,±9), then [tex]c=9.[/tex]
Note that
[tex]c^2=b^2+a^2,[/tex]
then
[tex]9^2=6^2+a^2,\\ \\a^2=81-36,\\ \\a^2=45.[/tex]
The equation of the hyperbola is
[tex]\dfrac{y^2}{36}-\dfrac{x^2}{45}=1.[/tex]
