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2009 Paper 1 Q8
D: 1500.0 B: 1484.0
circle tangent incircle triangle trigonometric identity tan 2α double angle formula distance from point to line coordinate geometry

  1. The equation of the circle \(C\) is \[ (x-2t)^2 +(y-t)^2 =t^2, \] where \(t\) is a positive number. Show that \(C\) touches the line \(y=0\,\). Let \(\alpha\) be the acute angle between the \(x\)-axis and the line joining the origin to the centre of \(C\). Show that \(\tan2\alpha=\frac43\) and deduce that \(C\) touches the line \(3y=4x\,\).
  2. Find the equation of the incircle of the triangle formed by the lines \(y=0\), \(3y=4x\) and \(4y+3x=15\,\). Note: The incircle of a triangle is the circle, lying totally inside the triangle, that touches all three sides.

Solution:

  1. This is a circle centre \((2t,t)\) with radius \(t\). Therefore it is exactly \(t\) away from the line \(y = 0\) so just touches that line. Not that \(\tan \alpha = \frac{t}{2t} = \frac12\) so \(\tan 2\alpha = \frac{2\tan \alpha}{1-\tan^2\alpha} = \frac{1}{1-\frac14} = \frac43\). Therefore the line \(y = \frac43x\) or \(3y = 4x\) is tangent to \(C\).
  2. Note that \(3y=4x\) and \(4y+3x=15\) are perpendicular, so this is a right-angled triangle with incenter \((2t,t)\) for some \(t\) and hypotenuse \(15\) We can find the third coordinate when \(3y-4x = 0\) and \(4y+3x = 15\) meet, ie \((\frac{9}{5}, \frac{12}5)\) The incentre lies on the bisector of the right angle at this point, which is the line through \((\frac{9}{5}, \frac{12}5)\) and \((\frac{15}{2}, 0)\), so \begin{align*} && \frac{2t-\frac{12}{5}}{t - \frac{9}{5}} &= \frac{-\frac{12}{5}}{\frac{15}2-\frac95} \\ \Rightarrow && \frac{10t-12}{5t-9} &= \frac{-24}{57} = -\frac{8}{19} \\ \Rightarrow && 190t - 12 \cdot 19 &= -40t + 72 \\ \Rightarrow && t &= 2 \end{align*} Therefore the center is \((4, 2)\) and the equation is \((x-4)^2+(y-2)^2=2^2\)

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2005 Paper 2 Q5
D: 1600.0 B: 1470.1
coordinate geometry right triangle incircle circumcircle tangent area optimisation inequality quadratic

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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1995 Paper 2 Q7
D: 1600.0 B: 1516.7
coordinate geometry triangle incircle Heron's formula area of triangle cosine rule semi-perimeter sphere proof

The diagram shows a circle, of radius \(r\) and centre \(I\), touching the three sides of a triangle \(ABC\). We write \(a\) for the length of \(BC\) and \(\alpha\) for the angle \(\angle BAC\) and so on. Let \(s=\frac{1}{2}\left(a+b+c\right)\) and let \(\triangle\) be the area of the triangle.

TikZ diagram
  1. By considering the area of the triangles \(AIB,\) \(BIC\) and \(CIA\), or otherwise, show that \(\Delta=rs\).
  2. By using the formula \(\Delta=\frac{1}{2}bc\sin\alpha\), show that \[ \Delta^{2}=\tfrac{1}{16}[4b^{2}c^{2}-\left(2bc\cos\alpha\right)^{2}]. \] Now use the formula \(a^{2}=b^{2}+c^{2}-2bc\cos\alpha\) to show that \[ \Delta^{2}=\tfrac{1}{16}[(a^{2}-\left(b-c\right)^{2})(\left(b+c\right)^{2}-a^{2})] \] and deduce that \[ \Delta=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}. \]
  3. A hole in the shape of the triangle \(ABC\) is cut in the top of a level table. A sphere of radius \(R\) rests in the hole. Find the height of the centre of the sphere above the level of the table top, expressing your answer in terms of \(a,b,c,s\) and \(R\).

Solution:

  1. \([AIB] = \frac12br\), \([BIC] = \frac12ar\), \([CIA] = \frac12 rc\), therefore \(\Delta = [AIB] +[BIC] + [CIA] = \frac12r(a+b+c) = sr\)
  2. \(\,\) \begin{align*} && \Delta &= \frac12 bc \sin \alpha \\ \Rightarrow && \Delta^2 &= \frac14 b^2c^2 \sin^2 \alpha \\ &&&= \frac14 \left (b^2c^2 - b^2c^2\cos^2 \alpha \right) \\ &&&= \frac1{16} \left (4b^2c^2 - (2bc\cos \alpha )^2\right) \\ \\ \Rightarrow && \Delta^2 &= \frac1{16} \left (4b^2c^2 - (b^2+c^2-a^2 )^2\right) \\ &&&= \frac1{16} (2bc-b^2-c^2+a^2)(2bc+b^2+c^2-a^2) \\ &&&= \frac{1}{16}(a^2-(b-c)^2)((b+c)^2-a^2) \\ &&&= \frac1{16}(a-b+c)(a+b-c)(b+c-a)(b+c+a) \\ &&&= (s - b)(s-c)(s-a)s \\ \Rightarrow && \Delta &= \sqrt{s(s-a)(s-b)(s-c)} \end{align*}
  3. We have the setting like this,
    TikZ diagram
    so \begin{align*} && h & = \sqrt{R^2-r^2} \\ &&&= \sqrt{R^2-\frac{\Delta^2}{s^2}} \\ &&&= \sqrt{R^2 - \frac{(s-a)(s-b)(s-c)}{s}} \end{align*}

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