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I am having a lot of trouble answering the following question as it is required to consider the value of t also. If t did not have to be considered, it would have been an easy ellipse.

Let an object move such that $$ \left\{ \begin{array}{cr} x =& \cos{2t} \\ y =& -\sin{t} \end{array} \right.\qquad \text{where} \quad 0 \leq t \leq \frac{5\pi}{4} $$ Sketch a graph of the path followed by the object.

I greatly appreciate your contributions. Thank you in advance

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Let $u=\sin t$, then since $\cos2t=1-2\sin^2t$, the curve can be reparametrised to $(1-u^2,-u)$ for $-\frac{\sqrt2}2\le u\le1$ (the range of $\sin t$ restricted to $0\le t\le\frac{5\pi}4$). This can then be implicitised as $x=1-2y^2$, so the curve is a parabola lying between the points obtained by substituting $t=-\frac{\sqrt2}2$ and $t=1$ into the parametric form (i.e. $(0,\sqrt2/2)$ and $(-1,-1)$. The rest of the sketch should be easy from here.

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  • $\begingroup$ Thanks, @ParclyTaxel ! $\endgroup$ – wendy Aug 12 at 9:24
  • $\begingroup$ @wendy Would you please accept my answer by clicking the tick left of it? $\endgroup$ – Parcly Taxel Aug 12 at 9:39
  • $\begingroup$ Hi, @ParclyTaxel , I didn't realise I hadn't accepted your answer! Sorry for not doing it sooner. $\endgroup$ – wendy Aug 13 at 11:38

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