2005 Paper 2 Q5

Year: 2005
Paper: 2
Question Number: 5

Course: LFM Pure
Section: Coordinate Geometry

Difficulty: 1600.0 Banger: 1470.1

coordinate geometry right triangle incircle circumcircle tangent area optimisation inequality quadratic

Problem

The angle $A$ of triangle $ABC$ is a right angle and the sides $BC$, $CA$ and $AB$ are of lengths $a$, $b$ and $c$, respectively. Each side of the triangle is tangent to the circle $S_1$ which is of radius $r$. Show that $2r = b+c-a$. Each vertex of the triangle lies on the circle~$S_2$. The ratio of the area of the region between~$S_1$ and the triangle to the area of $S_2$ is denoted by $R\,$. Show that $$ \pi R = -(\pi-1)q^2 + 2\pi q -(\pi+1) \;, $$ where $q=\dfrac{b+c}a\,$. Deduce that $$ R\le \frac1 {\pi( \pi - 1)} \;. $$

No solution available for this problem.

Rating Information

Difficulty Rating: 1600.0

Difficulty Comparisons: 0

Banger Rating: 1470.1

Banger Comparisons: 2

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Problem source

The angle $A$ of triangle $ABC$ is a right angle and the sides
$BC$, $CA$ and $AB$ are  of lengths $a$, $b$ and $c$,
respectively.  
Each side of the triangle is  tangent to the circle $S_1$ which is
of radius $r$.
Show that $2r = b+c-a$.

Each vertex of the triangle 
lies on the circle~$S_2$.
The ratio of  the area  of the  region between~$S_1$ 
and the triangle to the area of $S_2$ is denoted by $R\,$. 
Show that 
$$
\pi R = -(\pi-1)q^2 + 2\pi q -(\pi+1) \;,
$$
where  $q=\dfrac{b+c}a\,$.
Deduce that 
$$
R\le \frac1 {\pi( \pi - 1)} \;.
$$

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