Question

How do I rotate the axes of and then graph `11x^2+5.5y^2-22x+11y=0`?

Answer 1

There is no `xy` term, so it is not necessary to rotate the axes.

Explanation:

`11x^2+5.5y^2-22x+11y=0`

Complete the square to put this into standard form:

`(11x^2-22xcolor(white)"XX")+(5.5y^2+11ycolor(white)"XX")=0`

`11(x^2-2xcolor(white)"XXX")+5.5(y^2+2ycolor(white)"XXX")=0`

`11(x^2-2x+1)+5.5(y^2+2y+1)=0+11+5.5`

`11(x-1)^2+5.5(y+1)^2=16.5`

`(x-1)^2/(16.5/11)+(y+1)^2/(16.5/5.5)=1`

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

The graph is an ellipse with center #(1,-1) and endpoints of axes:

`(1+-sqrt3/2,-1)` and `(1, -1+-sqrt3)`.

graph{11x^2+5.5y^2-22x+11y=0 [-4.03, 7.07, -4.313, 1.237]}