Question

$y = \frac{(ax - b)(dx - c)}{(bx - a)(cx - d)}$

$$\begin{matrix} \Rightarrow & (ybc - ad)x^{2} - (bd + ac)(y - 1)x + (ady - bc) = 0 \\ \Rightarrow & (bd + ac)^{2}(y - 1)^{2} - 4(ybc - ad)(ady - bc) \geq 0 \\ \Rightarrow & y^{2}(bd - ac)^{2} + 2y\left( 2a^{2}d^{2} + 2b^{2}c^{2} - (bd + ac)^{2} \right) + (bd - ac)^{2} \geq 0\forall y \in R \\ \Rightarrow & \left( 2a^{2}d^{2} + 2b^{2}c^{2} - (bd + ac)^{2} \right) - (bd - ac)^{4} \leq 0 \\ \Rightarrow & 4\left( a^{2}d^{2} + b^{2}c^{2} - b^{2}d^{2} - a^{2}c^{2} \right)\left( a^{2}d^{2} + b^{2}c^{2} - 2abcd \right) \leq 0 \\ \Rightarrow & \left( a^{2} - b^{2} \right)\left( c^{2} - d^{2} \right)(ad - bc)^{2} \geq 0 \\ \Rightarrow & \left( a^{2} - b^{2} \right)\left( c^{2} - d^{2} \right) > 0 \end{matrix}$$