Question

Which of the following equations represents parametrically, parabolic profile ?

• A. $x=3cost;y=4sint$
• B. $x^2-2=-cost;y=4co{s}^2\frac{t}{2}$
• C. $\sqrt{x}=tant;\sqrt{y}=sect$
• D. $x=\sqrt{1-sint};y=sin\frac{t}{2}+cos\frac{t}{2}$

Answer 1

The correct option is B $x^2-2=-cost;y=4co{s}^2\frac{t}{2}$

Option$a)$

$x=3\cos t\Rightarrow \cos t=\frac{x}{3}$

$y=4\sin t\Rightarrow \sin t=\frac{y}{4}$

We know the identity, ${\sin }^{2}t+{\cos }^{2}t=1$

$\Rightarrow {\left(\frac{y}{4}\right)}^2+{\left(\frac{x}{3}\right)}^2=1$

$\Rightarrow \frac{y^2}{16}+\frac{x^2}{9}=1$

$\Rightarrow \frac{x^2}{9}+\frac{y^2}{16}=1$ represents an equation of ellipse.

Option$b)$

$x^2-2=-2\cos t$

$\Rightarrow x^2=2-2\cos t$

$\Rightarrow x^2=2(1-\cos t)$

$\Rightarrow x^2=2(1-(1-2{\sin }^2\frac{t}{2}))$

$\Rightarrow x^2=4{\sin }^2\frac{t}{2}$

We have $y=4{\cos }^2\frac{t}{2}$

$\Rightarrow {\cos }^2\frac{t}{2}=\frac{y}{4}$

We know the identity, ${\sin }^2\frac{t}{2}+{\cos }^2\frac{t}{2}=1$

$\Rightarrow \frac{x^2}{4}+\frac{y}{4}=1$

$\Rightarrow x^2=4-y$ represents a parabolic profile.

Option$c)$

$\sqrt{x}=\tan t,\sqrt{y}=\sec t$

We know that ${\sec }^{2}t-{\tan }^{2}t=1$

$\Rightarrow {\left(\sqrt{y}\right)}^2-{\left(\sqrt{x}\right)}^2=1$

$\Rightarrow y-x=1$ or $x-y+1=0$ is the equation of a straight line.

Option$d)$

$x=\sqrt{1-\sin t}=\sqrt{{\sin }^{2}t+{\cos }^{2}t-\sin t}$ using the identity ${\sin }^{2}t+{\cos }^{2}t=1$

$=\sqrt{{\sin }^{2}t+{\cos }^{2}t-2\sin \frac{t}{2}\cos \frac{t}{2}}$ using multiple angle formula $\sin t=2\sin \frac{t}{2}\cos \frac{t}{2}$

$=\sqrt{{(\sin \frac{t}{2}-\cos \frac{t}{2})}^2}$

$=|\sin \frac{t}{2}-\cos \frac{t}{2}|$

$\therefore x=\sin \frac{t}{2}-\cos \frac{t}{2}$ and $x=\sin \frac{t}{2}+\cos \frac{t}{2}$

We have $y=\sin \frac{t}{2}+\cos \frac{t}{2}$

$\Rightarrow x+y=\sin \frac{t}{2}-\cos \frac{t}{2}+\sin \frac{t}{2}+\cos \frac{t}{2}=2\sin \frac{t}{2}$

$\Rightarrow \sin \frac{t}{2}=\frac{y+x}{2}$

$\Rightarrow x-y=\sin \frac{t}{2}-\cos \frac{t}{2}-\sin \frac{t}{2}-\cos \frac{t}{2}=-2\cos \frac{t}{2}$

$\Rightarrow \cos \frac{t}{2}=\frac{y-x}{2}$

We have the identity

${\sin }^2\frac{t}{2}+{\cos }^2\frac{t}{2}=1$

$\Rightarrow {\left(\frac{y+x}{2}\right)}^2+{\left(\frac{y-x}{2}\right)}^2=1$

$\Rightarrow x^2+y^2+2xy+x^2+y^2-2xy=4$

$\Rightarrow x^2+y^2=2$ is the equation of the circle.