2005 Paper 1 Q2

Year: 2005
Paper: 1
Question Number: 2

Course: LFM Pure and Mechanics
Section: Differentiation from first principles

Difficulty: 1516.0 Banger: 1484.0

conics parabola tangent-normal parametric locus geometric proof differentiation from first principles

Problem

The point \(P\) has coordinates \(\l p^2 , 2p \r\) and the point \(Q\) has coordinates \(\l q^2 , 2q \r\), where \(p\) and~\(q\) are non-zero and \(p \neq q\). The curve \(C\) is given by \(y^2 = 4x\,\). The point \(R\) is the intersection of the tangent to \(C\) at \(P\) and the tangent to \(C\) at \(Q\). Show that \(R\) has coordinates \(\l pq , p+q \r\). The point \(S\) is the intersection of the normal to \(C\) at \(P\) and the normal to \(C\) at \(Q\). If \(p\) and \(q\) are such that \(\l 1 , 0 \r\) lies on the line \(PQ\), show that \(S\) has coordinates \(\l p^2 + q^2 + 1 , \, p+q \r\), and that the quadrilateral \(PSQR\) is a rectangle.

No solution available for this problem.

Rating Information

Difficulty Rating: 1516.0

Difficulty Comparisons: 1

Banger Rating: 1484.0

Banger Comparisons: 1

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Problem source

The point $P$ has coordinates $\l p^2  ,  2p \r$ 
and the point $Q$ has coordinates $\l q^2  ,  2q \r$, 
where $p$ and~$q$ are non-zero and $p \neq q$. 
The curve $C$ is given by  $y^2 = 4x\,$.
The point $R$ is the intersection of the tangent to $C$  at $P$ 
and the tangent 
to $C$ at $Q$. 
Show that $R$ has coordinates $\l pq ,  p+q \r$.

The point $S$ is the intersection of the normal to $C$ at $P$ 
and the normal to $C$ at $Q$. 
If $p$ and $q$ are such  that $\l 1  ,  0 \r$ 
lies on the line $PQ$, 
show that $S$ has coordinates $\l p^2 + q^2 + 1  , \, p+q \r$, 
and that the quadrilateral $PSQR$ is a rectangle.

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