1993 Paper 3 Q8

Year: 1993
Paper: 3
Question Number: 8

Course: LFM Pure and Mechanics
Section: Vectors

Difficulty: 1700.0 Banger: 1484.0

vectors 3D geometry plane equation square pyramid parametric proof parallelogram inequality

Problem

A square pyramid has its base vertices at the points $A$ $(a,0,0)$, $B$ $(0,a,0)$, $C$ $(-a,0,0)$ and $D$ $(0,-a,0)$, and its vertex at $E$ $(0,0,a)$. The point $P$ lies on $AE$ with $x$-coordinate $\lambda a$, where $0<\lambda<1$, and the point $Q$ lies on $CE$ with $x$-coordinate $-\mu a$, where $0<\mu<1$. The plane $BPQ$ cuts $DE$ at $R$ and the $y$-coordinate of $R$ is $-\gamma a$. Prove that $$ \gamma = {\lambda \mu \over \lambda + \mu - \lambda \mu}. $$ Show that the quadrilateral $BPRQ$ cannot be a parallelogram.

No solution available for this problem.

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Difficulty Rating: 1700.0

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Banger Rating: 1484.0

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Problem source

A square pyramid has its base vertices at the points $A$ $(a,0,0)$,
$B$ $(0,a,0)$, $C$ $(-a,0,0)$ and $D$ $(0,-a,0)$, and its vertex at 
$E$ $(0,0,a)$. The point $P$ lies  on $AE$ with $x$-coordinate $\lambda a$,
where $0<\lambda<1$, and the point $Q$ lies on $CE$ with $x$-coordinate
$-\mu a$, where $0<\mu<1$. The plane $BPQ$ cuts $DE$
at $R$ and the $y$-coordinate of $R$ is $-\gamma a$. Prove that 
$$
\gamma = {\lambda \mu \over \lambda + \mu - \lambda \mu}.
$$
Show that the quadrilateral $BPRQ$ cannot be a parallelogram.

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