Question

Rotate the axes to eliminate the xy-term in the equation. Then write the equation in standard form.

x^2 + 10xy + y^2 - 8 = 0

Answer 1

To solve this problem, we are going to rotate from the x-y plane to a new "u-v plane." Set x=u-v and y=u+v. Your equation then becomes \[{(u-v)}^2+10(u-v)(u+v)+{(u+v)}^2-8=0\] Which reduces to \[u^2-2uv+v^2+10u^2-10v^2+u^2+2uv+v^2-8=0\] which simplifies to \[12u^2-8v^2=8\] Dividing both sides by 8 gives \[\frac{3u^2}{2}-v^2=1\] which in standard form is just \[\frac{u^2}{\frac{2}{3}}-v^2=1\] Which is the equation of your rotated hyperbola in the uv plane.