Question

A hyperbola with a horizontal transverse axis contains the point at (4, 3). The equations of the asymptotes are y-x=1 and y+x=5. Write an equation for the hyperbola.

Answer 1

@campbell_st ?

Answer 2

hmm I assume you've covered hyperbolas?

Answer 3

Sort of

Answer 4

I know how to read it, just not quite sure how to write the equation myself

Answer 5

ok

well.... first off, let us notice that the hyperbola has a horizontal traverse axis
meaning is like

if it opens sideways, like so, it means it opens in the x-axis direction
and that means for the fractions, that the one with the "x" is the positive one
or \(\bf \cfrac{(x-{\color{brown}{ h}})^2}{{\color{purple}{ a}}^2}-\cfrac{(y-{\color{blue}{ k}})^2}{{\color{purple}{ b}}^2}=1
\qquad center\ ({\color{brown}{ h}},{\color{blue}{ k}})\qquad
vertices\ ({\color{brown}{ h}}\pm a, {\color{blue}{ k}})
\\ \quad \\
asymptotes\implies y={\color{blue}{ k}}\pm \cfrac{b}{a}(x-{\color{brown}{ h}})\)