2026 HCI P2 Q3

Junior College 2

11 marks

The curve $C$ has parametric equations

\[x=2e^t-1,\quad y=3e^{2t},\quad t\geq-1.\]

Sketch $C$, showing clearly the exact coordinates of any end points and axial intercepts.[2]
$P$ is a point on $C$ with parameter $p$. Show that the equation of the tangent at $P$ is \[y=ke^p(x-e^p+1),\] where $k$ is a constant to be determined.[4]
The tangent to the curve at $P\left(\dfrac12,\dfrac{27}{16}\right)$, meets the $x$-axis at point $A$. Find the area of the triangle $OAP$ where $O$ is the origin. Give your answer in exact form.[5]

Solution:

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Answer:(a) $y=\frac34(x+1)^2$, $x\geq\frac2e-1$; endpoint $(\frac2e-1,\frac3{e^2})$, $y$-intercept $(0,\frac34)$, no $x$-intercept. (b) $k=3$ (c) $\frac{27}{128}\text{ units}^2$.

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