로그곡선 위 넓이 비

네 점 $\mathrm{A}$, $\mathrm{Q}$, $\mathrm{P}$, $\mathrm{R}$의 좌표가 각각 $( 1 , 0 )$, $( a , 0 )$, $( a , b )$, $( 0 , b )$이므로 $\displaystyle ( \text{삼각형} \mathrm{AQP} \text{의} \text{넓이} ) = \frac{\mathrm{1}}{2} \overline{AQ} \cdot \overline{PQ}$ $\displaystyle = \frac{1}{2} ( a - 1 ) \cdot b$ $\displaystyle ( \text{사각형} \mathrm{OAPR} \text{의} \text{넓이} ) = \frac{\mathrm{1}}{2} ( \overline{OA} + \overline{PR} ) \cdot \overline{OR}$$\displaystyle = \frac{1}{2} ( 1 + a ) \cdot b$ $\displaystyle \frac{( \text{사각형} \mathrm{OAPR} \text{의} \text{넓이} )}{( \text{삼각형} \mathrm{AQP} \text{의} \text{넓이} )} = \frac{5}{4}$에서 $\displaystyle \frac{\frac{1}{2} ( 1 + a ) \cdot b}{\frac{1}{2} ( a - 1 ) \cdot b} = \frac{5}{4}$ $\therefore 5 a - 5 = 4 + 4 a \therefore a = 9 \cdots \cdots \text{㉠}$ ${ P} ( a , b )$는 $y = \log _{3} x$위에 있으므로 $b = \log _{3} 9 = 2$ ∴ $ab = 9 \cdot 2 = 18$