A curve $C$ has parametric equations $x=\frac{a}{t}$, $y=\ln {{t}^{a}}$, where $a$ is a

Question

A curve $C$ has parametric equations $x=\frac{a}{t}$, $y=\ln {{t}^{a}}$, where $a$ is a positive constant and $t>0$. Find the equations of the tangent and normal to the curve at the point $P$ with parameter $p$.[3] 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] Sketch curve $C$ and state, in terms of $a$, the coordinates of any point(s) where the curve crosses the axes.[1] 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]

Accepted Answer

(i) $y=-px+a+a\ln p$; $y=\frac{1}{p}x-\frac{a}{{{p}^{2}}}+a\ln p$ (ii) $\frac{{{a}^{2}}}{2p}\left( 1+\frac{1}{{{p}^{2}}} \right)$ units$^2$ (iii) $y=-x+a$; $-1<m<0$

2022 ACJC Promo Q8

2022 ACJC Promo Q8

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