Question

# Find an equation of the hyperbola satisfying the indicated properties. Vertices at

Transformation properties
Find an equation of the hyperbola satisfying the indicated properties.
Vertices at $$\displaystyle{\left({0},\ {1}\right)}$$ and $$\displaystyle{\left({0},\ -{1}\right)}$$. asymptotes are the lines $$\displaystyle{x}=\pm{3}{y}$$

2021-08-23
Step 1
Observe that only y-coordinate of vertices is changing, thus we must have an up-down hyperbola whose standard equation is given by:
$$\displaystyle{\frac{{{y}^{{{2}}}}}{{{a}^{{{2}}}}}}-{\frac{{{x}^{{{2}}}}}{{{b}^{{{2}}}}}}={1}$$
Step 2
Also note that in the case of up-down hyperbola, we have vertices $$\displaystyle{\left({0},\ {a}\right)}$$ and $$\displaystyle{\left({0},\ -{a}\right)},$$ thus comparing with the given vertices $$\displaystyle{\left({0},\ {1}\right)}$$ and $$\displaystyle{\left({0},\ −{1}\right)},$$ we have
$$\displaystyle{a}={1}$$
Step 3
Next recall that in this case, equation of the asymptotes is given by $$\displaystyle{x}=\pm{\frac{{{b}}}{{{a}}}}{y}$$, comparing this with the given asymptotes $$\displaystyle{x}=\pm{3}{y}$$, we get
$$\displaystyle{b}={3}$$
Step 4
Finally we plug the value $$\displaystyle{a}={1}$$ and $$\displaystyle{b}={3}$$ in the standard form of hyperbola to get the required equation as shown:
$$\displaystyle{\frac{{{y}^{{{2}}}}}{{{1}^{{{2}}}}}}-{\frac{{{x}^{{{2}}}}}{{{3}^{{{2}}}}}}={1}$$
$$\displaystyle{y}^{{{2}}}-{\frac{{{x}^{{{2}}}}}{{{9}}}}={1}$$
$$\displaystyle{y}^{{{2}}}-{\frac{{{x}^{{{2}}}}}{{{9}}}}={1}$$