If the normal to the rectangular hyperbola xy | vedclass

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(B) The parametric coordinates of a point on the rectangular hyperbola $xy = c^2$ are $(ct, c/t)$.The slope of the tangent at point $t$ is given by $\

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$-1$

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(B) The parametric coordinates of a point on the rectangular hyperbola $xy = c^2$ are $(ct, c/t)$.The slope of the tangent at point $t$ is given by $\frac{dy}{dx} = -\frac{1}{t^2}$.Therefore,the slope of the normal at point $t$ is $m_N = t^2$.The equation of the normal at point $t$ is $y - \frac{c}{t} = t^2(x - ct)$,which simplifies to $y - \frac{c}{t} = t^2x - c t^3$,or $y = t^2x - c t^3 + \frac{c}{t}$.Since this normal meets the curve again at $t_1$,the point $(ct_1, c/t_1)$ must satisfy the equation of the normal:$\frac{c}{t_1} = t^2(ct_1) - ct^3 + \frac{c}{t}$Dividing by $c$ (assuming $c 
eq 0$):$\frac{1}{t_1} = t^2 t_1 - t^3 + \frac{1}{t}$$\frac{1}{t_1} - \frac{1}{t} = t^2 t_1 - t^3$$\frac{t - t_1}{t t_1} = -t^2(t - t_1)$Since $t 
eq t_1$ for the normal to meet the curve at a different point,we can divide by $(t - t_1)$:$\frac{1}{t t_1} = -t^2$$1 = -t^3 t_1$$t^3 t_1 = -1$.

Column-$I$	Column-$II$
$A$. Circle	$P$. Locus of point $(h, k)$ such that the line $hx + ky = 1$ touches the circle $x^2 + y^2 = 4$
$B$. Parabola	$Q$. Point $z$ in the complex plane satisfies $\|z + 2\| - \|z - 2\| = \pm 3$
$C$. Hyperbola	$R$. Eccentricity of the conic lies in the interval $[1, \infty)$
	$S$. Point $z$ in the complex plane satisfies $Re(z + 1)^2 = \|z\|^2 + 1$

If the normal to the rectangular hyperbola $xy = c^2$ at the point $t$ meets the curve again at $t_1$,then $t^3 t_1$ has the value equal to

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