Question

Reduce the following equations to standard form, and then identify which conicoids they represent. Further, give a rough sketch of each.
i) x^2 +y^2 +2x−y−z+3 = 0
ii) 3y^2 +3z^2 +4x+3y+z = 9

Answer 1

$Given \ equation\ is \ x^2+y^2+2x-y-z+3\\ Completing \ the \ squares \ we \ get \ x^2+2x+1+y^2-2(\frac{1}{2})(y)+\frac{1}{4}-\frac{1}{4}-z-\frac{1}{4}+2=0\\ \Rightarrow\ (x+1)^2+(y-\frac{1}{2})^2+\frac{7}{4}-z=0\\ \Rightarrow \ z=(x+1)^2+(y-\frac{1}{2})^2+\frac{7}{4}\\ The\ graph\ of \ the \ given\ equation\ is ​ ​ ​$

$ii) \ 3y^2+3z^2+4x+3y+z=9\\ Completing\ the\ squares \ we\ get\\ 3(y+\frac{1}{2})^2+3(z+\frac{1}{6})^2+4x=(\sqrt\frac{59}{6})^2\\ The\ graph \ of \ the\ given\ curve\ is \\$