Mathematics - Area Under Curves Question with Solution | TestHub

The parallelogram formed by $y=ax\pm c^{\prime },y=bx\pm c^{\prime \prime }$ has area $\frac{{\left(c^{\prime }-c^{\prime \prime }\right)}^2}{|a-b|}$ in the appropriate combinations, giving $\frac{(c-d{)}^2}{|a-b|}=18$ and $\frac{(c+d{)}^2}{|a-b|}=72$. Dividing: $(c+d{)}^2=4(c-d{)}^2\Rightarrow c=3d$. Then $4d^2=18|a-b|$, i.e. $2d^2=9|a-b|$, whose smallest positive-integer solution is $|a-b|=2,d=3,c=9$; taking $a=3,b=1$ gives the least sum $3+1+9+3=16$.