Question

A particle moves in xy plane according to the law x = a sin ωt and y = a(1 - cos ωt), where a and w are constant. The particle traces

• (A) a straight line equally inclined to x-axis and y-axis
• (B) a parabola
• (C) a circle
• (D) None of these

Answer 1

The Correct Option is C

Explanation:
$x=a\sin \omega t\dots (i)$$y=a(1-\cos \omega t)\dots (ii)$$x^2=a^2{\sin }^2\omega t=a^2(1-{\cos }^2\omega t)$$x^2=a^2-a^2{\cos }^2\omega t$$a^2{\cos }^2\omega t=a^2-x^2$$y=a(1-\cos \omega t)=a-a\cos \omega t$$a-y=a\cos \omega t$$(a-y{)}^2=(a\cos \omega t{)}^2$$a^2+y^2-2ay=a^2{\cos }^2\omega t\dots (iv)$Put the value from equation (iii), $a^2+y^2-2ay=a^2-x^2$$x^2+y^2-2ay=0$This is the equation of a circle.