Pt. – Archived from the original on 2016-04-16. WORLD OF SUBWAYS Vol. 3: Keygen – Google Search.Q:

How many Points remain in P(L,E) for this problem?

Problem: Fix two distinct points A and B on a smooth quadratic
surface
S with a common tangent line (line AB) and consider the pencil
L (line A-B), E (line B-A). Each pencil consists of a real line and
the tangent line to S.
Show that there are infinitely many rays of the pencil L
passing through the point E such that the total length of the rays is
a rational multiple of the length of line E.
(This problem was included in the qualifying exam of the 7th grader, namely the ordinair)
I could not show this proof, so I’m trying to solve this problem by using s-t-computation, but I have not successfully. Could anyone tell me how to solve this problem?

A:

You can use two lemmas, $\angle A{LB}=\angle A{EA}=\frac\pi2$, $\angle B{EA}=\angle B{LE}=\frac\pi2$, and the fact that the sum of angles in a triangle is $2\pi$.

Let $H$ be the projection of $E$ onto $LS$. It is easily seen that $E\in LS$.
Define $h$ as $|\vec{HB}|$, and $Hx=hE$.
Define $\lambda=|\vec{EH}|$ and $\mu=|\vec{HE}|$; clearly, $\mu=|\vec{HB}|=h$.

$\angle H{LH}+\angle E{LE}=\frac\pi2+\frac\pi2=\pi$.

Therefore, $\mu-h=\lambda-h$; so, $h(1-\frac\lambda\mu)=0$.

Let $A’$ and $B’$ be the projections of $A$ and $B$ onto $LS$; they are the points whose norms are $a’$ and $b’$, so the lemma yields

\frac{1-a

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