Answer:
[tex]\frac{y^2}{144}-\frac{x^2}{64}=1[/tex]
Step-by-step explanation:
The equation of a hyperbola centered at the origin with vertices on the y-axis is given by: [tex]\frac{y^2}{a^2}-\frac{x^2}{b^2}=1[/tex]
The vertices of the hyperbola are the y-intercepts (0,12) and (0,-12)
This implies that:
[tex]2a=|12--12|[/tex]
[tex]2a=24[/tex]
[tex]a=12[/tex]
The asymptote equation of a hyperbola is given by:
[tex]y=\pm\frac{a}{b}x[/tex]
The given hyperbola has asymptote: [tex]y=\pm\frac{3}{2} x[/tex]
By comparison; [tex]\frac{a}{b}=\frac{3}{2}[/tex]
[tex]\implies \frac{12}{b}=\frac{12}{8}[/tex]
[tex]\implies b=8[/tex]
The required equation is:
[tex]\frac{y^2}{12^2}-\frac{x^2}{8^2}=1[/tex]
Or
[tex]\frac{y^2}{144}-\frac{x^2}{64}=1[/tex]
