두 포물선과 수평선

네 점 ${\mathrm{A}}$, ${\mathrm{B}}$, ${\mathrm{C}}$, ${\mathrm{D}}$의 $x$좌표를 각각 $\alpha _{1}$, $\beta _{1}$, $\alpha _{2}$, $\beta _{2}$라 하자. $\displaystyle {\overline{\mathrm{AB}}} = \beta _{1} - \alpha _{1}$, $\displaystyle {\overline{\mathrm{CD}}} = \beta _{2} - \alpha _{2}$ $\alpha _{1}$, $\beta _{1}$은 각각 $f \left( x \right) = k$의 해이므로 $\alpha _{1} + \beta _{1} = 2$, $\alpha _{1} \beta _{1} = 2 - k$ $\alpha _{2}$, $\beta _{2}$는 각각 $g \left( x \right) = k$의 해이므로 $\alpha _{2} + \beta _{2} = 8$, $\alpha _{2} \beta _{2} = 6 + k$ $\displaystyle {\overline{\mathrm{AB}}} = 2 {\overline{\mathrm{CD}}}$이므로 $\beta _{1} - \alpha _{1} = 2 \left( \beta _{2} - \alpha _{2} \right)$ $\left( \beta _{1} - \alpha _{1} \right) ^{2} = 4 \left( \beta _{2} - \alpha _{2} \right) ^{2}$ $\left( \alpha _{1} + \beta _{1} \right) ^{2} - 4 \alpha _{1} \beta _{1} = 4 \left\{ \left( \alpha _{2} + \beta _{2} \right) ^{2} - 4 \alpha _{2} \beta _{2} \right\}$ $2 ^{2} - 4 \left( 2 - k \right) = 4 \left\{ 8 ^{2} - 4 \left( 6 + k \right) \right\}$, $\displaystyle k = \frac{41}{5}$ 따라서 $10 k = 82$