The tangent to the hyperbola $xy = 1$ at the point $P$ intersects the $X$-axis at $T$ and the $Y$-axis at $T'$. The normal to the hyperbola at $P$ intersects the $X$-axis at $N$ and the $Y$-axis at $N'$. Let the areas of the triangles $PNT$ and $PN'T'$ are $\Delta$ and $\Delta'$ respectively. If $\frac{1}{\Delta} + \frac{1}{\Delta'}$ is constant for all positions of $P$ then the value of constant is ______.
Step-by-Step Solution
Key Concept: The sum of reciprocals of areas formed by tangent and normal is constant due to a harmonic property of parametric curves.
For the parametric curve, tangent at $P\left(t, \frac{1}{t}\right)$ is $x + t^2y = 2t$ with $T(2t, 0)$, and normal is $|a|^3x - ty = t^4 - 1$ with $N\left(t - \frac{1}{t^3}, 0\right)$. The areas of triangles $APT$ and $APT'N'$ give $\frac{1}{A} + \frac{1}{A'} = 2\frac{t^4+1}{1+t^4} = 2$, which is constant.
Correct Answer: 2