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Question

If rectangular hyperbola $$x^{2}- y^{2} = 4$$ is converted to $$xy = -c^{2}$$, then the equation of tangent to the hyperbola $$xy =- c^{2}$$ at point $$t, t$$ being a parameter, is :


A
xt+yt=±2
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B
xt+yt±42=0
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C
xtyt±22=0
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D
xtyt±42=0
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Solution

The correct option is C $$\dfrac{x}{t}-yt\pm 2\sqrt{2} =0$$
$$x^2-y^2=a^2$$ is converted into $$xy=-\dfrac{a^2}{2}$$, if axis is rotated by $$45^\circ$$ in clockwise direction. 
So in given equation,
$$c^2=\dfrac{a^2}{2}=\dfrac{4}{2}=2$$
$$c=\pm \sqrt2$$
Any parametric point on hyperbola $$=\left(ct,\dfrac{-c}{t}\right)$$
Equation of tangent $$xct-y\dfrac{c}{t}=-2c^2$$
$$\Rightarrow xt-\dfrac{y}{t}=-2c $$

$$ \Rightarrow xt-\dfrac{y}{t}\pm 2\sqrt2=0$$

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