Ellipse tangent intercepts and their midpoint · source reference

Printed question

The curve $C$ has parametric equations $x=4\cos\theta$, $y=3\sin\theta$, $0<\theta\le\pi$.

(a) Point $Q$ 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 $\theta$ at the instant when the coordinates of $Q$ is $(2,\frac{3\sqrt3}2)$. [3]

It is given that point $P$ also lies on $C$ and has parameter $p$ where $0<p<\pi/2$.

(b) Show that the equation of the tangent to $C$ at the point $P$ is given by $y=(-\frac34\cot p)x+3\operatorname{cosec}p$. [3]

(c) 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]

(d) Let $M$ be the mid-point of $AB$. Find a cartesian equation of the curve traced by $M$ as $p$ varies. [2]

[2023/ACJC/Promo/7]

Printed working

(a) $x=4\cos\theta$, $dx/d\theta=-4\sin\theta$.

$$\frac{d\theta}{dt}=\frac{d\theta}{dx}\frac{dx}{dt}=\frac{-1}{4\sin\theta}(15).$$

At $(2,3\sqrt3/2)$, $\theta=\pi/3$, so $d\theta/dt=-15/(4\sin(\pi/3))=-5\sqrt3/2=-15/(2\sqrt3)$.

(b) $dy/dx=\frac{3\cos\theta}{-4\sin\theta}=-\frac34\cot\theta$. At $P(4\cos p,3\sin p)$, $\theta=p$.

$$y-3\sin p=-\frac34\cot p(x-4\cos p),$$ $$y=-\frac34\cot p\,x+3\sin p+3\frac{\cos^2p}{\sin p}=-\frac34\cot p\,x+3\frac{\sin^2p+\cos^2p}{\sin p}=-\frac34\cot p\,x+3\operatorname{cosec}p.$$

(c) $A=(4/\cos p,0)$, $B=(0,3/\sin p)$. Area $OAB=\frac12(4/\cos p)(3/\sin p)=12/\sin2p=6/(\sin p\cos p)$.

(d) $M=(2/\cos p,3/(2\sin p))$. Parametric equations $x=2/\cos p$, $y=3/(2\sin p)$. Since $\sin^2p+\cos^2p=1$, Cartesian equation $\left(\frac3{2y}\right)^2+\left(\frac2x\right)^2=1$. Given the parameter interval, retain $x>2$, $y>3/2$.

Teacher notes

Given $dx/dt$, to find $d\theta/dt$ apply chain rule: $d\theta/dt=(d\theta/dx)(dx/dt)$.

To find the tangent at point $P$, where $\theta=p$, remember to substitute $p$ into $dy/dx$.

To obtain a Cartesian equation, eliminate variable $p$ using the identity relating $\cos p$ and $\sin p$, namely $\sin^2p+\cos^2p=1$.

Answer key

(a) $-5\sqrt3/2$. (b) $y=(-3\cot p/4)x+3\operatorname{cosec}p$. (c) $12/\sin2p$. (d) $(3/(2y))^2+(2/x)^2=1$.

Native diagrams