Question

1. sketch the following quadric surface.

\\(\frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{6} = 1\\)

Explanation:

Step1: Identify the Surface Type

The given equation is \(\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{6} = 1\). The general form of an ellipsoid is \(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\), where \(a\), \(b\), \(c\) are the semi - axes lengths along the \(x\), \(y\), \(z\) axes respectively. Here, \(a = \sqrt{4}=2\), \(b=\sqrt{9} = 3\), \(c=\sqrt{6}\approx2.45\). So this is an ellipsoid.

Step2: Find Intercepts

• \(x\) - intercept: Set \(y = 0\) and \(z=0\) in the equation. We get \(\frac{x^2}{4}=1\), so \(x^2 = 4\) and \(x=\pm2\). The \(x\) - intercepts are \((2,0,0)\) and \((- 2,0,0)\).
• \(y\) - intercept: Set \(x = 0\) and \(z = 0\) in the equation. We get \(\frac{y^2}{9}=1\), so \(y^2=9\) and \(y = \pm3\). The \(y\) - intercepts are \((0,3,0)\) and \((0, - 3,0)\).
• \(z\) - intercept: Set \(x = 0\) and \(y=0\) in the equation. We get \(\frac{z^2}{6}=1\), so \(z^2 = 6\) and \(z=\pm\sqrt{6}\approx\pm2.45\). The \(z\) - intercepts are \((0,0,\sqrt{6})\) and \((0,0,-\sqrt{6})\).

Step3: Analyze Cross - Sections

• Cross - section with \(xy\) - plane (\(z = 0\)): The equation becomes \(\frac{x^2}{4}+\frac{y^2}{9}=1\), which is an ellipse with semi - major axis \(b = 3\) (along \(y\) - axis) and semi - minor axis \(a = 2\) (along \(x\) - axis).
• Cross - section with \(xz\) - plane (\(y = 0\)): The equation becomes \(\frac{x^2}{4}+\frac{z^2}{6}=1\), which is an ellipse with semi - axes \(a = 2\) (along \(x\) - axis) and \(c=\sqrt{6}\) (along \(z\) - axis).
• Cross - section with \(yz\) - plane (\(x = 0\)): The equation becomes \(\frac{y^2}{9}+\frac{z^2}{6}=1\), which is an ellipse with semi - major axis \(b = 3\) (along \(y\) - axis) and semi - axis \(c=\sqrt{6}\) (along \(z\) - axis).

Step4: Sketching the Ellipsoid

1. Plot the intercepts \((\pm2,0,0)\), \((0,\pm3,0)\) and \((0,0,\pm\sqrt{6})\) in a 3 - D coordinate system.
2. Draw the ellipses in the \(xy\), \(xz\) and \(yz\) planes as analyzed above.
3. Connect these ellipses smoothly to form the ellipsoid, making sure to show the curvature and the relative lengths of the semi - axes.

Answer:

The given quadric surface \(\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{6}=1\) is an ellipsoid with semi - axes \(a = 2\) (along \(x\) - axis), \(b = 3\) (along \(y\) - axis) and \(c=\sqrt{6}\) (along \(z\) - axis). The sketch involves plotting the intercepts and the cross - sectional ellipses and connecting them to form the ellipsoid.