Find the equation of the hyperbola passing through the points $(1,5)$ and $(\frac{-5}{2},\frac{11}{5})$ with the vertical asymptote $x= \frac{5}{4}$.

Question

Given: vertical asymptote: $x = \frac{5}{4}$, Passing through points: $(1,5)$ and $\left(\frac{-5}{2},\frac{11}{5}\right)$.

Because we are given vertical asymptote , we should have a rectangular hyperbola:$$y =\dfrac{a}{x-h} +k$$

However, I don't know how to find $a$ and $k$ (horizontal asymptote. Is it supposed to be $\frac{5}{4}$?).

Answer 1

Hint: vertical asymptote at $x=\frac{5}{4}$ means that your denominator becomes $0$ when you have $x=\frac{5}{4}$. Together with the two given points, you have three equations and three unknowns, solve them!