Quadric Surfaces — Question 3

Question 3 – Solution

Strategy Complete squares; a single negative term with right side $-1$ is more clearly read after multiplying by $-1$.

Standard form Completing squares gives $$9(x+2)^2-4(y-1)^2+9(z-1)^2=-4,$$ hence $$\boxed{\frac{(y-1)^2}{1}-\frac{(x+2)^2}{4/9}-\frac{(z-1)^2}{4/9}=1}.$$ This is a hyperboloid of two sheets with axis parallel to $y$.

Vertices and coordinate traces The vertices are $\boxed{(-2,0,1),(-2,2,1)}$. All three coordinate planes intersect the surface: $y=0$ contains the first vertex, while $x=0$ and $z=0$ give hyperbolas. The plane $y=1$ does not intersect the surface, but it is not a coordinate plane.