Question

Given x(t) = 3sin(1000$\pi$t ) and y(t) = 5cos(1000$\pi$t + $\frac{\pi }{4}$ ) The x-y plot will be

• A. a circle
• B. a multi-loop closed curve
• C. a hypernbola
• D. an ellipse

Answer 1

The correct option is D an ellipse

Let $\theta =1000\pi t$ then $\frac{x}{3}=sin\theta$ ...(1)

and $\frac{y}{5}=cos(\theta +\frac{\pi }{4})=\frac{1}{\sqrt{2}}(cos\theta -sin\theta )$..(2)

Solving (1) and (2)

$\frac{y}{5}=\frac{1}{\sqrt{2}}(cos\theta -\frac{x}{3})$

$\frac{y\sqrt{2}}{5}+\frac{x}{3}=cos\theta$ ...(3)

So, $si{n}^2\theta +co{s}^2\theta =1$

$\Rightarrow \frac{x^2}{9}+{(\frac{x}{3}+\frac{y\sqrt{2}}{5})}^2=1$

$\frac{2x^2}{9}+\frac{2\sqrt{2}}{15}xy+\frac{2}{25}{y}^2-1=0$ ....(4)

On comparison with $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$

c = 0

$a=\frac{2}{9},h=\frac{\sqrt{2}}{15},b=\frac{2}{25}$

$\because h^2-ab=\frac{-2}{225}<0$

and $\Delta =\left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|=-\frac{2}{225}\ne 0$

So, (4) represents and Ellipse