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

1 Answer
Jul 8, 2017

Answer:

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]}