The curve $C$ has parametric equations $x=4\cos heta $, $y=3\sin heta $, $0Point $Q$ lies on $C$

Question

The curve $C$ has parametric equations $x=4\cos 	heta $, $y=3\sin 	heta $, $0Point $Q$ lies on $C$ and is moving along $C$, such that its x coordinate is increasing at the constant rate of $15$ units per second. Determine the exact rate of change of $	heta $ at the instant when the coordinates of $Q$ is $\left( 2,\frac{3\sqrt{3}}{2} ight)$. [3] It is given that point $P$ also lies on $C$ and has parameter $p$ where $0 Show that the equation of the tangent to $C$ at the point $P$, is given by $y=\left( -\frac{3}{4}\cot p ight)x+3\,\mathrm{cosec}\,p$. [3] The tangent to the curve at $P$ meets the $x$-axis at the point $A$ and the $y$-axis at the point $B$. Find the area of the triangle $OAB$. [2] Let $M$ be the mid-point of $AB$. Find a cartesian equation of the curve traced by $M$ as $p$ varies. [2]

Accepted Answer

(a) $\frac{\mathrm d\theta}{\mathrm dt}=-\frac{5\sqrt3}{2}\text{ rad s}^{-1}$. (b) $y=-\frac34(\cot p)x+3\csc p$. (c) $6\sec p\csc p\text{ units}^2$. (d) $\frac4{x^2}+\frac9{4y^2}=1$, $x>2$, $y>\frac32$.

2023 ACJC Promo Q7

2023 ACJC Promo Q7

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