Dolly Robinson

2021-08-22

Find an equation of the hyperbola satisfying the indicated properties.

Vertices at$(0,\text{}1)$ and $(0,\text{}-1)$ . asymptotes are the lines $x=\pm 3y$

Vertices at

ottcomn

Skilled2021-08-23Added 97 answers

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:

$\frac{{y}^{2}}{{a}^{2}}-\frac{{x}^{2}}{{b}^{2}}=1$

Step 2

Also note that in the case of up-down hyperbola, we have vertices$(0,\text{}a)$ and $(0,\text{}-a),$ thus comparing with the given vertices $(0,\text{}1)$ and $(0,\text{}-1),$ we have

$a=1$

Step 3

Next recall that in this case, equation of the asymptotes is given by$x=\pm \frac{b}{a}y$ , comparing this with the given asymptotes $x=\pm 3y$ , we get

$b=3$

Step 4

Finally we plug the value$a=1$ and $b=3$ in the standard form of hyperbola to get the required equation as shown:

$\frac{{y}^{2}}{{1}^{2}}-\frac{{x}^{2}}{{3}^{2}}=1$

${y}^{2}-\frac{{x}^{2}}{9}=1$

Answer:

${y}^{2}-\frac{{x}^{2}}{9}=1$

Observe that only y-coordinate of vertices is changing, thus we must have an up-down hyperbola whose standard equation is given by:

Step 2

Also note that in the case of up-down hyperbola, we have vertices

Step 3

Next recall that in this case, equation of the asymptotes is given by

Step 4

Finally we plug the value

Answer:

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