Question

complete the sentence below. for the hyperbola $\frac{y^{2}}{16}-\frac{x^{2}}{81}=1$, the asymptotes are and for the hyperbola $\frac{y^{2}}{16}-\frac{x^{2}}{81}=1$, the asymptotes are $y=-4x$ and $y = 4x$. $y=-\frac{9}{4}x$ and $y=\frac{9}{4}x$. $y=-\frac{4}{9}x$ and $y=\frac{4}{9}x$. $y=-9x$ and $y = 9x$.

Explanation:

Step1: Recall the formula for asymptotes of a hyperbola

For a hyperbola of the form \(\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}} = 1\), the equations of the asymptotes are \(y=\pm\frac{a}{b}x\).

Step2: Identify \(a\) and \(b\)

Given \(\frac{y^{2}}{16}-\frac{x^{2}}{81}=1\), we have \(a^{2} = 16\) so \(a = 4\), and \(b^{2}=81\) so \(b = 9\).

Step3: Substitute \(a\) and \(b\) into the asymptote formula

Substituting \(a = 4\) and \(b = 9\) into \(y=\pm\frac{a}{b}x\), we get \(y=\pm\frac{4}{9}x\).

Answer:

\(y =-\frac{4}{9}x\) and \(y=\frac{4}{9}x\)