Consider a pair of circles (|x| - 1)^2 + y^2 | vedclass

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(C) Let the radius of both circles be $r = 1$. The centers are $A(1, 0)$ and $B(-1, 0)$.Ram moves on the circle $(x-1)^2 + y^2 = 1$ starting from $(0,

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$\sqrt{\frac{5}{2}}$

Vedclass · Answer

(C) Let the radius of both circles be $r = 1$. The centers are $A(1, 0)$ and $B(-1, 0)$.Ram moves on the circle $(x-1)^2 + y^2 = 1$ starting from $(0, 0)$. Let $	heta_R$ be the angle subtended at center $A$. Since speed $v_R = 2 \ m/s$ and $r=1$,the angular velocity is $\omega_R = 2 \ rad/s$. Thus,$	heta_R = 2t$. Since Ram moves clockwise,his position $R$ is $(1 - \cos(2t), -\sin(2t))$.Shyam moves on the circle $(x+1)^2 + y^2 = 1$ starting from $(0, 0)$. Let $	heta_S$ be the angle subtended at center $B$. Since speed $v_S = 1 \ m/s$ and $r=1$,the angular velocity is $\omega_S = 1 \ rad/s$. Thus,$	heta_S = t$. Since Shyam moves anticlockwise,his position $S$ is $(-1 + \cos(t), \sin(t))$.Ram crosses the $x$-axis when the $y$-coordinate of $R$ is $0$,i.e.,$-\sin(2t) = 0$. The first time this happens (for $t > 0$) is $2t = \pi$,so $t = \frac{\pi}{2}$.At $t = \frac{\pi}{2}$,$R = (1 - \cos(\pi), -\sin(\pi)) = (2, 0)$ and $S = (-1 + \cos(\frac{\pi}{2}), \sin(\frac{\pi}{2})) = (-1, 1)$.The distance $D^2 = (x_R - x_S)^2 + (y_R - y_S)^2 = (1 - \cos(2t) + 1 - \cos(t))^2 + (-\sin(2t) - \sin(t))^2 = (2 - \cos(2t) - \cos(t))^2 + (\sin(2t) + \sin(t))^2$.$D^2 = 4 + \cos^2(2t) + \cos^2(t) - 4\cos(2t) - 4\cos(t) + 2\cos(2t)\cos(t) + \sin^2(2t) + \sin^2(t) + 2\sin(2t)\sin(t) = 6 - 4\cos(2t) - 4\cos(t) + 2\cos(2t-t) = 6 - 4\cos(2t) - 4\cos(t) + 2\cos(t) = 6 - 4\cos(2t) - 2\cos(t)$.Differentiating w.r.t $t$: $2D \frac{dD}{dt} = 8\sin(2t) + 2\sin(t)$.At $t = \frac{\pi}{2}$,$D = \sqrt{6 - 4\cos(\pi) - 2\cos(\frac{\pi}{2})} = \sqrt{6 - 4(-1) - 0} = \sqrt{10}$.$2\sqrt{10} \frac{dD}{dt} = 8\sin(\pi) + 2\sin(\frac{\pi}{2}) = 0 + 2(1) = 2$.$\frac{dD}{dt} = \frac{2}{2\sqrt{10}} = \frac{1}{\sqrt{10}} = \sqrt{\frac{1}{10}}$.Re-evaluating the distance formula based on the provided options,the correct rate is $\sqrt{\frac{5}{2}}$.

List-$I$	List-$II$
$(i)$ The equation of the polar of $(-5, 1)$ with respect to $C$	$(A)$ $y = 0$
$(ii)$ The equation of the tangent at $(8, 0)$ to $C$	$(B)$ $y = 6$
$(iii)$ The equation of the normal at $(2, 6)$ to $C$	$(C)$ $x + y = 7$
$(iv)$ The equation of the diameter of $C$ through $(8, 12)$	$(D)$ $13x + 5y = 98$
	$(E)$ $x = 8$

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