Let $a$ and $b$ be positive real numbers such that $a > 1$ and $b < a$. Let $P$ be a point in the first quadrant that lies on the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$. 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?

$a>1, \quad 0<b<a, \quad P \text{ on } \dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1$ $\text{tangent at } P \text{ passes through } (1,0)$ $\text{normal at } P \text{ cuts equal intercepts on the axes}$ $\text{Find: which are TRUE? } (1)\ 1<e<\sqrt{2}, \ (2)\ \sqrt{2}<e<2$ $(3)\ \Delta=a^{4}, \quad (4)\ \Delta=b^{4}$ $1.\ \text{tangent at } P: \dfrac{xx_{1}}{a^{2}}-\dfrac{yy_{1}}{b^{2}}=1$ $2.\ (1,0) \text{ on tangent} \Rightarrow x_{1}=a^{2}$ $3.\ \text{normal: equal intercepts} \Rightarrow b^{2}; \ \text{find } e,\ \Delta$ $\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1 \quad (\text{representative } a=2,\ b=\sqrt{3})$ $\text{Tangent at } P(x_{1},y_{1}): \dfrac{xx_{1}}{a^{2}}-\dfrac{yy_{1}}{b^{2}}=1$ $(1,0): \dfrac{x_{1}}{a^{2}}=1 \Rightarrow x_{1}=a^{2}$ $\dfrac{x_{1}^{2}}{a^{2}}-\dfrac{y_{1}^{2}}{b^{2}}=1 \Rightarrow y_{1}^{2}=b^{2}(a^{2}-1)$ $y_{1}=b\sqrt{a^{2}-1}$ $m_{\text{tan}}=\dfrac{b^{2}x_{1}}{a^{2}y_{1}}=\dfrac{b}{\sqrt{a^{2}-1}}$ $m_{\text{norm}}=-\dfrac{\sqrt{a^{2}-1}}{b}$ $\text{equal intercepts} \Rightarrow |m_{\text{norm}}|=1$ $\dfrac{\sqrt{a^{2}-1}}{b}=1 \Rightarrow b^{2}=a^{2}-1$ $e^{2}=1+\dfrac{b^{2}}{a^{2}}=2-\dfrac{1}{a^{2}}$ $a>1 \Rightarrow 1<e^{2}<2 \Rightarrow 1<e<\sqrt{2}$ $\therefore (1)\ \text{TRUE}, \quad (2)\ \text{FALSE}$ $\text{tangent: } x-y=1$ $\text{normal: } y=-x+2a^{2}-1$ $A(1,0), \ B(2a^{2}-1,0), \ C=P(a^{2},a^{2}-1)$ $\Delta=\dfrac{1}{2}\cdot 2(a^{2}-1)\cdot(a^{2}-1)=(a^{2}-1)^{2}=b^{4}$ $\therefore (4)\ \text{TRUE}, \quad (3)\ \text{FALSE}$ $\boxed{(1)\ \text{and}\ (4)\ \text{are TRUE}}$