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

i) $x^2+y^2+2x-y-z+3=0$

then we group variables. We get equation. $(x+1)^2+(y-1/2)^2+7/4=z$

Finish formul is $(x+1)^2+(y-1/2)^2=z-7/4$

This formula is the formula of elliptic paraboloid (we then can see from standard form of elliptic paraboloid, $z-z_0=A(x-x_0)^2+B(y-y_0)^2$ , where the coefficients A and B have the same sings )

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ii) $3y^2 +3z^2 +4x+3y+z = 9$

Then, also we do group variables. We get the formula. $3(y+1/2)^2-3/4+3(z+1/6)^2-1/12=9-4x$$3(y+1/2)^2+3(z+1/6)^2=49/6-4x$ All numbers and variable x on the right side.

This is also elliptic paraboloid, but this elliptic paraboloid going on x-os.

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