this is from the appendix 4 (the conic sections)in spivak calculus book.
i spend the whole day tried to understand what spivak means in the third picture ("we can make things a lot ....."),but without any reslut.
so please can you explain what exactly he means ?
Taking equation of cone and general plane as
$$ \dfrac{z^2}{x^2+y^2}= C^2 $$
and
$$ \dfrac{x}{a}+ \dfrac{y}{b}+ \dfrac{z}{c}=1$$ the orthogonal projection of intersection on $x-y$ plane is a conic section whose principal axes are not parallel to $x-y$ axes.
Let $ c \to \infty,~b =-ma~$ we have a single straight line projection parallel to the $x-y$ plane of ellipse or hyperbola section.
$$ y+ m x= m a $$
whose principal axes are parallel to $x$ and $y$ axes.
If the intersection plane is rotated by $45^{\circ}$ around z-axis, then
$$ x^2+y^2-x y=1 $$
a typical conic section projection given below is obtained,
but whose axes are not parallel to $x-y$ axes.
By everything Spivak means the cone and intersection plane together.. need to be rotated about z-axis to bring about the alignment.