1. sketch the following quadric surface.\n$$\\frac{y^{2}}{9}+z^{2}=1$$
1. sketch the following quadric surface.\n$$\\frac{y^{2}}{9}+z^{2}=1$$
Answer
Explanation:
Step1: Identify the surface type
The equation $\frac{y^2}{9} + z^2 = 1$ lacks the variable $x$, indicating a cylinder.
Step2: Analyze the cross-section
In the $yz$-plane, the equation represents an ellipse with semi-axes $a=3$ (along $y$) and $b=1$ (along $z$).
Step3: Extend along the missing axis
Since $x$ is independent, the elliptical cross-section is translated infinitely along the $x$-axis.
Step4: Describe the final shape
The surface is an elliptic cylinder centered on the $x$-axis, extending from $-\infty$ to $+\infty$.
Answer:
The quadric surface is an elliptic cylinder opening along the $x$-axis. To sketch it, draw an ellipse in the $yz$-plane centered at $(0,0,0)$ with vertices at $(0, \pm3, 0)$ and $(0, 0, \pm1)$, then extend this shape parallel to the $x$-axis.