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(C) Let the point on the hyperbola $xy = c^2$ be $P(ct, c/t)$.The equation of the tangent at $P$ is $\frac{x}{ct} + \frac{y}{c/t} = 2$,which simplifie

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(C) Let the point on the hyperbola $xy = c^2$ be $P(ct, c/t)$.The equation of the tangent at $P$ is $\frac{x}{ct} + \frac{y}{c/t} = 2$,which simplifies to $\frac{x}{t} + ty = 2c$.The $x$-intercept $a_1 = 2ct$ and the $y$-intercept $b_1 = 2c/t$.The equation of the normal at $P$ is $y - \frac{c}{t} = t^2(x - ct)$,which simplifies to $t^2x - y = ct^3 - c/t$.The $x$-intercept $a_2 = \frac{ct^3 - c/t}{t^2} = ct - \frac{c}{t^3}$.The $y$-intercept $b_2 = -(ct^3 - c/t) = \frac{c}{t} - ct^3$.Now,calculate $a_1a_2 + b_1b_2$:$a_1a_2 = (2ct)(ct - \frac{c}{t^3}) = 2c^2t^2 - \frac{2c^2}{t^2}$.$b_1b_2 = (\frac{2c}{t})(\frac{c}{t} - ct^3) = \frac{2c^2}{t^2} - 2c^2t^2$.Adding these gives $a_1a_2 + b_1b_2 = (2c^2t^2 - \frac{2c^2}{t^2}) + (\frac{2c^2}{t^2} - 2c^2t^2) = 0$.

If the tangent and normal to a rectangular hyperbola $xy = c^2$ at a variable point cut off intercepts $a_1, a_2$ on the $x$-axis and $b_1, b_2$ on the $y$-axis,then $(a_1a_2 + b_1b_2)$ is

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