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(D) Given the curve $y = 	an(	an^{-1} x) = x$.Since the circle passes through the point $S(1, 0)$ and is tangent to the line $L: x - y = 0$,the locu

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both $(A)$ and $(B)$

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(D) Given the curve $y = 	an(	an^{-1} x) = x$.Since the circle passes through the point $S(1, 0)$ and is tangent to the line $L: x - y = 0$,the locus of the centre $C(h, k)$ is a parabola where $S(1, 0)$ is the focus and $L: x - y = 0$ is the directrix.The distance from the focus $S(1, 0)$ to the directrix $x - y = 0$ is $2a = \frac{|1 - 0|}{\sqrt{1^2 + (-1)^2}} = \frac{1}{\sqrt{2}}$.The axis of the parabola is the line passing through the focus $(1, 0)$ and perpendicular to the directrix $x - y = 0$. The slope of the directrix is $1$,so the slope of the axis is $-1$.The equation of the axis is $y - 0 = -1(x - 1)$,which simplifies to $x + y = 1$.The intersection of the axis $x + y = 1$ and the directrix $x - y = 0$ is the point $Z(1/2, 1/2)$.The vertex $V$ is the midpoint of the focus $S(1, 0)$ and the point $Z(1/2, 1/2)$.$V = \left(\frac{1 + 1/2}{2}, \frac{0 + 1/2}{2}ight) = (3/4, 1/4)$.Since both $(A)$ and $(B)$ are correct,the answer is $(D)$.

$A$ variable circle is described to pass through the point $(1, 0)$ and is tangent to the curve $y = \tan(\tan^{-1} x)$. The locus of the centre of the circle is a parabola whose :

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