Question

Convert `9x^2-25y^2-54x-50y-119=0` in to standard equation of a hyperbola?

Answer 1

Please see below.

Explanation:

`9x^2-25y^2-54x-50y-119=0` can be written as

`9(x^2-6x)-25(y^2+2y)-119=0`
(here we have grouped `x` and `y` terms together)

or `9(x^2-6x+9-9)-25(y^2+2y+1-1)-119=0`
(here we have added and suntracted constant terms to make it a square)

or `9(x^2-6x+9)-25(y^2+2y+1)-81-119+25=0`
(here we have kept complete squares and moved extra constant terms out)

or `9(x-3)^2-25(y+1)^2=175`
(here written them as squares)

or `(9(x-3)^2)/175-(25(y+1)^2)/175=1`.
(and finally divided by `175` to get desired form and result.)

or `(x-3)^2/(sqrt(175/9))^2-(y+1)^2/7=1`.

or `(x-3)^2/((5sqrt7)/3)^2-(y+1)^2/(sqrt7)^2=1`.