1991 Paper 3 Q4

Year: 1991
Paper: 3
Question Number: 4

Course: LFM Pure
Section: 3x3 Matrices

Difficulty: 1700.0 Banger: 1516.0

3x3 matrices vectors straight line in 3D lines of sight parametric simultaneous equations 3D geometry intersection

Problem

The point \(P\) moves on a straight line in three-dimensional space. The position of \(P\) is observed from the points \(O_{1}(0,0,0)\) and \(O_{2}(8a,0,0).\) At times \(t=t_{1}\) and \(t=t_{1}'\), the lines of sight from \(O_{1}\) are along the lines \[ \frac{x}{2}=\frac{z}{3},y=0\quad\mbox{ and }\quad x=0,\frac{y}{3}=\frac{z}{4} \] respectively. At times \(t=t_{2}\) and \(t=t_{2}'\), the lines of sight from \(O_{2}\) are \[ \frac{x-8a}{-3}=\frac{y}{1}=\frac{z}{3}\quad\mbox{ and }\quad\frac{x-8a}{-4}=\frac{y}{2}=\frac{z}{5} \] respectively. Find an equation or equations for the path of \(P\).

No solution available for this problem.

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Difficulty Rating: 1700.0

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Banger Rating: 1516.0

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Problem source

The point $P$ moves on a straight line in three-dimensional space.
The position of $P$ is observed from the points $O_{1}(0,0,0)$ and
$O_{2}(8a,0,0).$ At times $t=t_{1}$ and $t=t_{1}'$, the lines of
sight from $O_{1}$ are along the lines 
\[
\frac{x}{2}=\frac{z}{3},y=0\quad\mbox{ and }\quad x=0,\frac{y}{3}=\frac{z}{4}
\]
respectively. At times $t=t_{2}$ and $t=t_{2}'$, the lines of sight
from $O_{2}$ are 
\[
\frac{x-8a}{-3}=\frac{y}{1}=\frac{z}{3}\quad\mbox{ and }\quad\frac{x-8a}{-4}=\frac{y}{2}=\frac{z}{5}
\]
respectively. Find an equation or equations for the path of $P$.

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