1994 Paper 3 Q3

Year: 1994
Paper: 3
Question Number: 3

Course: LFM Pure and Mechanics
Section: Vectors

Difficulty: 1700.0 Banger: 1516.0

vectors sphere plane circle intersection line of intersection 3D geometry geometric proof

Problem

Describe geometrically the possible intersections of a plane with a sphere. Let \(P_{1}\) and \(P_{2}\) be the planes with equations \begin{alignat*}{1} 3x-y-1 & =0,\\ x-y+1 & =0, \end{alignat*} respectively, and let \(S_{1}\) and \(S_{2}\) be the spheres with equations \begin{alignat*}{1} x^{2}+y^{2}+z^{2} & =7,\\ x^{2}+y^{2}+z^{2}-6y-4z+10 & =0, \end{alignat*} respectively. Let \(C_{1}\) be the intersection of \(P_{1}\) and \(S_{1},\) let \(C_{2}\) be the intersection of \(P_{2}\) and \(S_{2}\) and let \(L\) be the intersection of \(P_{1}\) and \(P_{2}.\) Find the points where \(L\) meets each of \(S_{1}\) and \(S_{2}.\) Determine, giving your reasons, whether the circles \(C_{1}\) and \(C_{2}\) are linked.

No solution available for this problem.

Rating Information

Difficulty Rating: 1700.0

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Banger Rating: 1516.0

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Problem source

Describe geometrically the possible intersections of a plane with
a sphere. 

Let $P_{1}$ and $P_{2}$ be the planes with equations 
\begin{alignat*}{1}
3x-y-1 & =0,\\
x-y+1 & =0,
\end{alignat*}
respectively, and let $S_{1}$ and $S_{2}$ be the spheres with equations
\begin{alignat*}{1}
x^{2}+y^{2}+z^{2} & =7,\\
x^{2}+y^{2}+z^{2}-6y-4z+10 & =0,
\end{alignat*}
respectively. Let $C_{1}$ be the intersection of $P_{1}$ and $S_{1},$
let $C_{2}$ be the intersection of $P_{2}$ and $S_{2}$ and let
$L$ be the intersection of $P_{1}$ and $P_{2}.$ Find the points
where $L$ meets each of $S_{1}$ and $S_{2}.$ Determine, giving
your reasons, whether the circles $C_{1}$ and $C_{2}$ are linked.

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