Question 170
In the Cartesian plane, let $\gamma_1$ and $\gamma_2$ be the graphs of the quadratic functions $f_1(x) = p_1x^2+q_1x+r_1$ and $f_2(x) = p_2x^2+q_2x+r_2$, where $p_1 \gt 0 \gt p_2$. The graphs $\gamma_1$ and $\gamma_2$ cross at distinct points $A$ and $B$. The four tangents to $\gamma_1$ and $\gamma_2$ at $A$ and $B$ form a convex quadrangle which has an inscribed circle. Prove that the graphs $\gamma_1$ and $\gamma_2$ have the same axis of symmetry.