2004 Paper 3 Q4

Year: 2004
Paper: 3
Question Number: 4

Course: LFM Pure
Section: Trigonometry 2

Difficulty: 1700.0 Banger: 1609.8

trigonometry geometric series isosceles triangle tangent circles ratio summation optimisation inequality

Problem

The triangle \(OAB\) is isosceles, with \(OA = OB\) and angle \(AOB = 2 \alpha\) where \(0< \alpha < {\pi \over 2}\,\). The semi-circle \(\mathrm{C}_0\) has its centre at the midpoint of the base \(AB\) of the triangle, and the sides \(OA\) and \(OB\) of the triangle are both tangent to the semi-circle. \(\mathrm{C}_1, \mathrm{C}_2, \mathrm{C}_3, \ldots\) are circles such that \(\mathrm{C}_n\) is tangent to \(\mathrm{C}_{n-1}\) and to sides \(OA\) and \(OB\) of the triangle. Let \(r_n\) be the radius of \(\mathrm{C}_n\,\). Show that \[ \frac{r_{n+1}}{r_n} = \frac{1-\sin\alpha}{1+\sin\alpha}\;. \] Let \(S\) be the total area of the semi-circle \(\mathrm{C}_0\) and the circles \(\mathrm{C}_1\), \(\mathrm{C}_2\), \(\mathrm{C}_3\), \(\ldots\;\). Show that \[ S = {1 + \sin^2 \alpha \over 4 \sin \alpha} \, \pi r_0^2 \;. \] Show that there are values of \(\alpha\) for which \(S\) is more than four fifths of the area of triangle~\(OAB\).

No solution available for this problem.

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Difficulty Rating: 1700.0

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Banger Rating: 1609.8

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Problem source

The triangle $OAB$ is isosceles, 
with $OA = OB$ and angle $AOB = 2 \alpha$ where $0< \alpha < {\pi \over 2}\,$. 
The semi-circle $\mathrm{C}_0$ has its centre at the midpoint of the base $AB$ of the triangle, 
and the sides $OA$ and $OB$ of the triangle are both tangent to the semi-circle. 
$\mathrm{C}_1, \mathrm{C}_2, \mathrm{C}_3, \ldots$ 
are circles such that $\mathrm{C}_n$ is tangent to $\mathrm{C}_{n-1}$ 
and to sides $OA$ and $OB$ of the triangle. 
Let $r_n$ be the radius of $\mathrm{C}_n\,$. Show that
\[
\frac{r_{n+1}}{r_n} = \frac{1-\sin\alpha}{1+\sin\alpha}\;.
\]
Let $S$ be the total area of the semi-circle $\mathrm{C}_0$ and the 
circles  $\mathrm{C}_1$, $\mathrm{C}_2$, $\mathrm{C}_3$, $\ldots\;$. 
Show that
\[
S = {1 + \sin^2 \alpha  \over 4 \sin \alpha} \, \pi r_0^2 \;.
\]
Show that there are values of $\alpha$ for which $S$ is more than four fifths 
of the area of triangle~$OAB$.

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