Chapter 16

Find the equation of the cylinder, whose guiding curve is \(x^{2}+z^{2}-4 x-2 z+4=0\), \(y=0\) and whose axis contains the point \((0,3,0)\). Find also the area of the section of the cylinder by a plane parallel to the \(x z\) plane.

Prove that the equation of the cylinder with generators parallel to \(z\) -axis and passing through the curve \(a x^{2}+b y^{2}=2 c z, l x+m y+n z=p\) is \(n\left(a x^{2}+b y^{2}\right)+\) \(2 c(l x+m y)-2 p c=0 .\)

A cylinder cuts the plane \(z=0\) with curve \(x^{2}+\frac{y^{2}}{4}=\frac{1}{4}\) and has its axis parallel to \(3 x=-6 y=2 z\). Find its equation.

A straight line is always parallel to the \(y z\) plane and intersects the curves \(x^{2}+y^{2}=a^{2}, z=0\) and \(x^{2}=a z, y=0\). Prove that it generates the surface \(x^{4} y^{2}=\left(x^{2}-a z\right)^{2}\left(a^{2}-x^{2}\right) .\)

Find the equation of the right circular cylinder of radius 2 whose axis passes through \((1,2,3)\) and has direction cosines proportional \(2,-3,6\).

Find the equation of the right circular cylinder of radius 1 with axis as \(\frac{x-1}{2}=\frac{y}{3}=\frac{z-3}{1}\)