Let \( a \) and \( b \) be positive real numbers such that \( a \)
\( \mathrm{P} \) 1. and \( ba \). Let \( P \) be a point in the first quadrant
W that lies on the hyperbola \( x^{2} / a^{2}-y^{2} / b^{2}=1 \).
Suppose the tangent to the hyperbola at \( P \) passes through the point \( (1,0) \), and suppose the normal to the hyperbola at \( P \) cuts off equal intercepts on the coordinate axes. Let \( \Delta \) denote the area of the triangle formed by the tangent at \( P \), the normal at \( P \) and the x-axis. If \( e \) denotes the eccentricity of the hyperbola, then which of the following statements is/are TRUE?
(1) \( 1e\sqrt{2} \)
(2) \( \sqrt{2}\mathrm{e}2 \)
(3) \( \Delta=a^{4} \)
(4) \( \Delta=b^{4} \)
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