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Prove that the equations \begin{align*} a(x) \equiv a_0x^3+a_1x^2+a_2x+a_3=0 \\ \text{and} \quad b(x) \equiv b_0x^3+b_1x^2+b_2x+b_3=0 \end{align*} will have a common root if \[ \Delta \equiv \begin{vmatrix} a_0 & a_1 & a_2 & a_3 & 0 & 0 \\ 0 & a_0 & a_1 & a_2 & a_3 & 0 \\ 0 & 0 & a_0 & a_1 & a_2 & a_3 \\ 0 & 0 & b_0 & b_1 & b_2 & b_3 \\ 0 & b_0 & b_1 & b_2 & b_3 & 0 \\ b_0 & b_1 & b_2 & b_3 & 0 & 0 \end{vmatrix} = 0. \] If \begin{align*} a(x) &= (x-\lambda)(a_0'x^2+a_1'x+a_2') \\ b(x) &= (x-\lambda)(b_0'x^2+b_1'x+b_2') \end{align*} and \[ \Delta_1 \equiv \begin{vmatrix} a_0 & a_1 & a_2 & a_3 \\ 0 & a_0 & a_1 & a_2 \\ 0 & b_0 & b_1 & b_2 \\ b_0 & b_1 & b_2 & b_3 \end{vmatrix} \] prove that \[ \Delta_1 \times \begin{vmatrix} 1 & \lambda & \lambda^2 & \lambda^3 \\ 0 & 1 & \lambda & \lambda^2 \\ 0 & 0 & 1 & \lambda \\ 0 & 0 & 0 & 1 \end{vmatrix} = \begin{vmatrix} a_0' & a_1' & a_2' & 0 \\ 0 & a_0' & a_1' & a_2' \\ 0 & b_0' & b_1' & b_2' \\ b_0' & b_1' & b_2' & 0 \end{vmatrix} \] and hence that the equations \(a(x)=0, b(x)=0\) will have two common roots if \(\Delta=0\) and \(\Delta_1=0\).

Prove that a circle can be drawn through the four points of intersection of two parabolas whose axes are at right angles. Show that the point of intersection of the axes of the two parabolas bisects the join of the centre of this circle to that of the rectangular hyperbola through the same four points.