Question

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

Transformation properties
ANSWERED
asked 2021-08-22
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 \(\displaystyle{x}=\pm{3}{y}\)

Expert Answers (1)

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:
\(\displaystyle{\frac{{{y}^{{{2}}}}}{{{1}^{{{2}}}}}}-{\frac{{{x}^{{{2}}}}}{{{3}^{{{2}}}}}}={1}\)
\(\displaystyle{y}^{{{2}}}-{\frac{{{x}^{{{2}}}}}{{{9}}}}={1}\)
Answer:
\(\displaystyle{y}^{{{2}}}-{\frac{{{x}^{{{2}}}}}{{{9}}}}={1}\)
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