2022 ACJC Promo Q8
A curve $C$ has parametric equations $x=\frac{a}{t}$, $y=\ln {{t}^{a}}$, where $a$ is a positive constant and $t>0$.
(i)
Find the equations of the tangent and normal to the curve at the point $P$ with parameter $p$.
[3]
(ii)
The tangent at $P$ and the normal at $P$ meet the $y$-axis at point $A$ and point $B$ respectively. Find the area of triangle $APB$ in terms of $a$ and $p$.
[2]
(iii)
Sketch curve $C$ and state, in terms of $a$, the coordinates of any point(s) where the curve crosses the axes.
[1]
(iv)
State the equation of the tangent to $C$ when $p=1$. Hence state the range of values of $m$ such that the line $y=mx+a$ cuts curve $C$ at two points.
[2]
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