The curve $C$ has parametric equations $x={{t}^{2}}$, $y={{t}^{2}}-t$. The point $P$ on the curve ha

Question

The curve $C$ has parametric equations $x={{t}^{2}}$, $y={{t}^{2}}-t$. The point $P$ on the curve has parameter $p$. Show that the equation of the tangent at $P$ is $2py=\left( 2p-1 \right)x-{{p}^{2}}$. [3] If the tangent at $P$ passes through the point $\left( 4,\frac{3}{2} \right)$, use a non-calculator method to find the possible coordinates of $P$. [2] The tangent at $P$ meets the line $y=x$ at $A$ and the $y$-axis at $B$. Show that the area of triangle $OAB$ is $\frac{1}{4}{{p}^{3}}$, where $O$ is the origin. [3] If the point $P$ moves along the curve $C$ such that its $x$-coordinate is increasing at a constant rate of $2$ units per second, find the rate of increase of the area of triangle $OAB$ when $P$ is at $\left( 16,12 \right)$. [4]

Accepted Answer

(i) $2py=(2p-1)x-p^2$ (ii) $P=(1,0)$ or $P=(16,12)$ (iii) $\mathcal A=\frac14p^3$ (iv) $3\text{ units}^2/\text{s}$

2013 NYJC P2 Q4

2013 NYJC P2 Q4

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