Quadric Surfaces — Question 1

Question 1 – Solution

Strategy Complete the square separately in $x,y,z$, then divide by the resulting constant.

Standard form Completing squares gives $$4(x-1)^2+(y+3)^2+9(z-2)^2=45,$$ so $$\boxed{\frac{(x-1)^2}{45/4}+\frac{(y+3)^2}{45}+\frac{(z-2)^2}{5}=1}.$$

Geometry This is an ellipsoid centered at $\boxed{(1,-3,2)}$ with semiaxes $3\sqrt 5/2$, $3\sqrt 5$, and $\sqrt 5$. Its axial extreme points are $$(1\pm 3\sqrt 5/2,-3,2),\quad(1,-3\pm 3\sqrt 5,2),\quad(1,-3,2\pm\sqrt 5).$$

Verification Substitution of any listed endpoint makes exactly one normalized square equal to 1.