A curve $C$ has parametric equations $x=3{{u}^{2}}$, $y=6u$. Find the equation of the normal to the

Question

A curve $C$ has parametric equations $x=3{{u}^{2}}$, $y=6u$. Find the equation of the normal to the curve at the point $P\left( 3{{p}^{2}},\ 6p \right)$, where $p$ is a non-zero constant. [2] The normal meets the $x$-axis at $Q$ and the $y$-axis at $R$. Find the coordinates of $Q$ and of $R$. [2] Find two possible expressions for the area of the triangle bounded by the normal and the axes in terms of $p$, stating the range of values of $p$ in each case. [3] Given that $p$ is positive and increasing at a rate of $2$ units/s, find the rate of change of the area of the triangle in terms of $p$. [2]

Accepted Answer

(a) $y=-px+3p^3+6p$. (b) $Q=(3p^2+6,0), R=(0,3p^3+6p)$. (c) $A=\frac92p(p^2+2)^2$ for $p>0$, and $A=-\frac92p(p^2+2)^2$ for $p<0$. (d) $45p^4+108p^2+36\text{ units}^2\text{/s}$.

2024 DHS Promo Q7

2024 DHS Promo Q7

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