Question

Which one of the following equations represent parametric equation to a parabolic curve?

• A. $x=3\cos t;y=4\sin t$
• B. $x^2-2=2\cos t;y=4{\cos }^2\frac{t}{2}$
• C. $\sqrt{x}=\tan t;\sqrt{y}=\sec t$
• D. $x=\sqrt{1-\sin t};y=\sin \frac{t}{2}+\cos \frac{t}{2}$

Answer 1

Answer: (B)

$x=3\cos t,y=4\sin t$
Eliminating $t$, we have
$\frac{x^2}{9}+\frac{y^2}{16}=1$
which is an ellipse. Therefore,
$x^2-2=2\cos t$ and $y=4{\cos }^2\frac{t}{2}$
or $y=2(1+2\cos t)$
and $y=2\left(1+\frac{x^2-2}{2}\right)$
which is a parabola.
$\sqrt{x}=\tan t;\sqrt{y}=\sec t$
Eliminating $t$, we have
$y-x=1$
which is a straight line.
$x=\sqrt{1-\sin t}$
$y=\sin \frac{t}{2}+\cos \frac{t}{2}$
Eliminating $t$, we have
$x^2+y^2=1-\sin t+1+\sin t=2$
which is a circle.