Question 881785
This is a ellipse.  The Distance Formula gives you a starting equation which with lengthy steps, and great attention to details, you can simplify.


{{{sqrt((x+2)^2+(y-2)^2)+sqrt((x-8)^2+(y-2)^2)=12}}}
You will want to subtract one of the radicals from both sides before squaring both sides, or your next steps can be extremely inconvenient.  You will also need to square a second time.


Let me skip MOST of the steps here, and I will bring to the first form of the quadratic, conic section equation:


{{{11x^2-66x+36y^2=153}}}, requiring completing the square for x;
{{{11(x^2-6x+9)+36y^2=153+99}}}
{{{highlight_green(11(x-3)^2+36y^2=162)}}}
Not yet in standard form, but you can finish doing that.