Question

How do use the method of translation of axes to sketch the curve `9x^2-4y^2-36x-24y-36=0`?

Answer 1

Please see below.

Explanation:

`9x^2-4y^2-36x-24y-36=0` can be written as

`9(x^2-4x+4)-36-4(y^2+6y+9)+36=36`

or `9(x-2)^2-4(y+3)^2=36`

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

Hence, let us move `(0,0)` to `(2,-3)`, i.e. `x->x'+2` and `y->y'-3`, which makes `x=2` as new `y`-axis and `y=-3` as new `x`-axis and our equation becomes

`(x'^2)/2^2-(y'^2)/3^2=1`, the equation of a hyperbola.

graph{9x^2-4y^2-36x-24y-36=0 [-8.59, 11.41, -7.96, 2.04]}

graph{x^2/4-y^2/9=1 [-10.59, 9.41, -5.2, 4.8]}