Question

Where is the curve x^(2)y-xy-x^(2)+10x-12y-21=0 symmetric?

Answer 1

first solve for y
\[y =f(x)= \frac{x^2-10x+21}{x^2 -x-12}\]
curve is symmetric when
f(x) = f(-x)
\[f(-x) = \frac{x^2 +10x+21}{x^2 +x-12}\]
Factor and equate the two functions
\[\frac{(x-7)(x-3)}{(x-4)(x+3)} = \frac{(x+7)(x+3)}{(x+4)(x-3)}\]
Unfortunately nothing cancels...
cross multiply
\[x^4-9x^3-x^2+141x-252 = x^4+9x^3-x^2-141x-252\]
\[18x^3 -282x = 0\]
\[6x(3x^2 -47) = 0\]
\[x = 0, \pm \sqrt{\frac{47}{3}}\]