Question

Which one of the following equations parametrically represents a parabolic profile ?

• A. $x=3\cos t,y=4\sin t$
• B. $x^2=-\cos t,y=4{\cos }^2\frac{t}{2}$
• 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)

$x^2=-\cos t,y=4{\cos }^2\frac{t}{2}$

option (A) $x=3cost:y=4sint$
eliminating 't' we get, $\frac{x^2}{9}+\frac{y^2}{16}=1$ Clearly this is an ellipse.
option (B) $x^2=-cost:y=4co{s}^2\frac{t}{2}=2(1+\cos t)$
eliminating 't' we get, $x^2=-(\frac{y}{2}-1)$ Clearly this is parabola.
option (C) $\sqrt{x}=tant\sqrt{y}=sect$
eliminating 't' we get, $y-x=1$ Clearly this is a straight line
option (D) $x=\sqrt{1-sint}:y=sin\frac{t}{2}+cos\frac{t}{2}$
$\Rightarrow x^2=1-\sin t,y^2=1+\sin t$
eliminating 't' we get, $x^2+y^2=2$ Clearly this is a circle.