포물선 초점 현

$\displaystyle \mathrm{\overline{BI}} = \mathit{3} k$라 하면 $\displaystyle \frac{\mathrm{\overline{BJ}}}{\overline{\mathrm{BI}}} = \frac{2 \sqrt{15}}{3}$에서 $\displaystyle \mathrm{\overline{BJ}} = \mathit{2} \sqrt{15} k$ 포물선의 정의에 의하여 $\displaystyle \mathrm{\overline{BI}} = \overline{BF} = \mathit{3} k$, $\displaystyle \mathrm{\overline{AF}} = \overline{AH} = 8 \sqrt{5} - 3 \mathit{k}$ $\displaystyle \mathrm{\overline{JH}} = \mathit{3} k$이므로 $\displaystyle \mathrm{\overline{AJ}} = \mathit{8} \sqrt{5} - 6 k$ 직각삼각형 $\mathrm{ABJ}$에서 $\displaystyle \left( 8 \sqrt{5} \right) ^{2} = \left( 2 \sqrt{15} k \right) ^{2} + \left( 8 \sqrt{5} - 6 k \right) ^{2}$ $\displaystyle 96 k ^{2} - 96 \sqrt{5} k = 0$, $\displaystyle 96 k \left( k - \sqrt{5} \right) = 0$ $k > 0$이므로 $\displaystyle k = \sqrt{5}$ 따라서 $\displaystyle \mathrm{\overline{AJ}} = 2 \sqrt{5}$, $\displaystyle \mathrm{\overline{BJ}} = 10 \sqrt{3}$, $\displaystyle \mathrm{\overline{AH}} = 5 \sqrt{5}$이고 $\triangle \mathrm{ABJ}$󰁀$\triangle \mathrm{ACH}$이므로 $\displaystyle \mathrm{\overline{AJ}} : \overline{JB} = \overline{AH} : \overline{HC}$ $\displaystyle 2 \sqrt{5} : 10 \sqrt{3} = 5 \sqrt{5} : \mathrm{\overline{HC}}$ $\therefore$$\displaystyle \mathrm{\overline{HC}} = 25 \sqrt{3}$