Question

1. sketch the following quadric surface.

\\(\frac{y^2}{9} + z^2 = 1\\)

Explanation:

Step1: Identify the type of quadric surface

The given equation is \(\frac{y^{2}}{9}+z^{2}=1\). Notice that there is no \(x\)-term, which means the surface is a cylinder (a cylindrical surface) extending along the \(x\)-axis. The equation in the \(y - z\) plane is an ellipse (since \(\frac{y^{2}}{3^{2}}+z^{2}=1\) is the standard form of an ellipse \(\frac{y^{2}}{a^{2}}+\frac{z^{2}}{b^{2}} = 1\) with \(a = 3\) and \(b=1\)).

Step2: Analyze the cross - sections

• Cross - section parallel to \(y - z\) plane (\(x=\text{constant}\)): For any fixed value of \(x\), the equation \(\frac{y^{2}}{9}+z^{2}=1\) represents an ellipse in the plane \(x = x_0\) with semi - major axis \(a = 3\) (along the \(y\)-axis) and semi - minor axis \(b = 1\) (along the \(z\)-axis).
• Cross - section parallel to \(x - y\) plane (\(z = 0\)): Substitute \(z = 0\) into the equation, we get \(\frac{y^{2}}{9}=1\), so \(y=\pm3\). This is a pair of vertical lines (in the \(x - y\) plane) at \(y = 3\) and \(y=- 3\) extending along the \(x\)-axis.
• Cross - section parallel to \(x - z\) plane (\(y = 0\)): Substitute \(y = 0\) into the equation, we get \(z^{2}=1\), so \(z=\pm1\). This is a pair of horizontal lines (in the \(x - z\) plane) at \(z = 1\) and \(z = - 1\) extending along the \(x\)-axis.

Step3: Sketch the surface

1. First, draw the ellipse \(\frac{y^{2}}{9}+z^{2}=1\) in the \(y - z\) plane. The ellipse has vertices at \((y,z)=(3,0)\), \((- 3,0)\), \((0,1)\) and \((0, - 1)\).
2. Then, since the surface is a cylinder along the \(x\)-axis, extend this ellipse infinitely in both the positive and negative \(x\)-directions. The surface is an elliptic cylinder with the generating curve (the ellipse \(\frac{y^{2}}{9}+z^{2}=1\)) in the \(y - z\) plane and the axis of the cylinder along the \(x\)-axis.

Answer:

The quadric surface \(\frac{y^{2}}{9}+z^{2}=1\) is an elliptic cylinder extending along the \(x\)-axis. The cross - section in the \(y - z\) plane is the ellipse \(\frac{y^{2}}{3^{2}}+z^{2}=1\), and it extends infinitely along the \(x\)-axis. To sketch it, draw the ellipse \(\frac{y^{2}}{9}+z^{2}=1\) in the \(y - z\) plane and then extend it parallel to the \(x\)-axis.