Standard +0.8 This question requires implicit differentiation of an exponential function, finding the tangent equation, determining both axis intercepts, and calculating a triangle area. While the techniques are standard C3/C4 content, the combination of exponential implicit differentiation with the multi-step geometric application and algebraic manipulation needed for exact form makes it moderately challenging, above average difficulty.
10. The curve \(C\) satisfies the equation
$$x \mathrm { e } ^ { 5 - 2 y } - y = 0 \quad x > 0 , \quad y > 0$$
The point \(P\) with coordinates ( \(2 \mathrm { e } ^ { - 1 } , 2\) ) lies on \(C\).
The tangent to \(C\) at \(P\) cuts the \(x\)-axis at the point \(A\) and cuts the \(y\)-axis at the point \(B\).
Given that \(O\) is the origin, find the exact area of triangle \(O A B\), giving your answer in its simplest form.
\includegraphics[max width=\textwidth, alt={}]{a377da06-a968-438c-bec2-ae55283dae47-35_4_21_127_2042} L
10. The curve $C$ satisfies the equation
$$x \mathrm { e } ^ { 5 - 2 y } - y = 0 \quad x > 0 , \quad y > 0$$
The point $P$ with coordinates ( $2 \mathrm { e } ^ { - 1 } , 2$ ) lies on $C$.\\
The tangent to $C$ at $P$ cuts the $x$-axis at the point $A$ and cuts the $y$-axis at the point $B$.\\
Given that $O$ is the origin, find the exact area of triangle $O A B$, giving your answer in its simplest form.\\
\includegraphics[max width=\textwidth, alt={}]{a377da06-a968-438c-bec2-ae55283dae47-35_4_21_127_2042} L\\
\hfill \mbox{\textit{Edexcel C34 2018 Q10 [7]}}