91Ó°ÊÓ

Find the equation of the cone whose vertex is at the point \((1,1,0)\) and whose guiding curve is \(x^{2}+z^{2}=4, y=0\).

A variable plane is parallel to the plane \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=0\) and meets the axes in \(A, B\) and \(C\). Prove that, the circle \(A B C\) lies on the cone \(y z\left(\frac{b}{c}+\frac{c}{a}\right)+z x\left(\frac{c}{a}+\frac{a}{c}\right)+x y\left(\frac{a}{b}+\frac{b}{a}\right)=0 .\)

i) Find the equation of the quadric cone which passes through the three coordinate axes and three mutually perpendicular lines \(\frac{x}{2}=\frac{y}{1}=\frac{z}{-1}, \frac{x}{1}=\frac{y}{3}=\frac{z}{5}, \frac{x}{8}=\frac{y}{-11}=\frac{z}{5} .\) ii) Prove that the equation of the cone whose vertex is \((0,0,0)\) and the base curve \(z=k, f(x, y)=0\) is \(f\left(\frac{x k}{z}, \frac{y k}{z}\right)=0\), where \(f(x, y)=a x^{2}+2 h x y+\) \(b y^{2}+2 g x+2 f y+c=0\)