I am posting three pictures from my text book regarding the derivation of the equation of cone which has the 6 normals from a point on a conicoid as its generators
I have posted the derivation from the very definition of normal , since the text uses equation derived from previous proofs . I have the following questions
1.While deriving the equation of the Quadric cone how did the author eliminate $r $ from the equations $\frac{af/l}{1+ar}=\frac{bg/m}{1+br}=\frac{ch/n}{1+cr} $(I have marked it with three questions marks)
2.Why is it necessary to even study about the number of normals drawn from a given point to a conicoid and the equation of the cone having these normals as generators
Set: $A=af/l$, $B=bg/m$, $C=ch/n$, and rewrite your equations as $$ {1+ar\over A}={1+br\over B}={1+cr\over C}. $$ To eliminate $r$, use the first equation to find $$ r={A-B\over aB-bA} $$ and plug that into the second equation: $$ C\left(1+b{A-B\over aB-bA}\right)=B\left(1+c{A-B\over aB-bA}\right). $$ Simplify to get the stated result.