Quadric Surfaces — Question 1

Question 1 – Solution

Strategy Group like variables and complete each square.

Step 1: Complete squares Write $$4(x^2-2x)+(y^2+6y)+9(z^2-4z)+4=0.$$ Using $x^2-2x=(x-1)^2-1$, $y^2+6y=(y+3)^2-9$, and $z^2-4z=(z-2)^2-4$ gives $$4(x-1)^2+(y+3)^2+9(z-2)^2=45.$$ Divide by $45$: $$\boxed{\frac{(x-1)^2}{45/4}+\frac{(y+3)^2}{45}+\frac{(z-2)^2}{5}=1}.$$

Step 2: Geometry This is an ellipsoid centered at $\boxed{(1,-3,2)}$ with semiaxes $3\sqrt 5/2$, $3\sqrt 5$, and $\sqrt 5$. Its six axis endpoints relative to the center are obtained by adding and subtracting those lengths in the corresponding coordinate.

Step 3: Principal traces Through the center, the traces are $$x=1:\ \frac{(y+3)^2}{45}+\frac{(z-2)^2}{5}=1,\quad y=-3:\ \frac{(x-1)^2}{45/4}+\frac{(z-2)^2}{5}=1,$$ $$z=2:\ \frac{(x-1)^2}{45/4}+\frac{(y+3)^2}{45}=1.$$

Step 4: Axis intercepts Set the other two coordinates to zero in the original equation. This gives $$\boxed{(1,0,0)},\qquad \boxed{(0,-3\pm\sqrt 5,0)},\qquad \boxed{(0,0,2\pm 4\sqrt 2/3)}.$$ Substitution verifies each intercept and trace.