Area with optimization or parameters

Find area as a function of a variable or parameter, often involving maximization/minimization or showing area expressions in terms of coordinates on a line

2 questions · Standard +0.3

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CAIE P1 2013 November Q6
7 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{d5f66324-e1fc-40e1-98e7-625187e24d3d-3_465_663_1160_740} In the diagram, \(S\) is the point ( 0,12 ) and \(T\) is the point ( 16,0 ). The point \(Q\) lies on \(S T\), between \(S\) and \(T\), and has coordinates \(( x , y )\). The points \(P\) and \(R\) lie on the \(x\)-axis and \(y\)-axis respectively and \(O P Q R\) is a rectangle.
  1. Show that the area, \(A\), of the rectangle \(O P Q R\) is given by \(A = 12 x - \frac { 3 } { 4 } x ^ { 2 }\).
  2. Given that \(x\) can vary, find the stationary value of \(A\) and determine its nature.
CAIE P1 2015 June Q6
7 marks Standard +0.3
The line with gradient \(-2\) passing through the point \(P(3t, 2t)\) intersects the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).
  1. Find the area of triangle \(AOB\) in terms of \(t\). [3]
The line through \(P\) perpendicular to \(AB\) intersects the \(x\)-axis at \(C\).
  1. Show that the mid-point of \(PC\) lies on the line \(y = x\). [4]