Mathematics - Application of Derivative Question with Solution | TestHub

Every member of the family passes through (1,2). Let the line be $\frac{x}{a}+\frac{y}{b}=1$ with $a,b>0$ and $\frac{1}{a}+\frac{2}{b}=1$. Minimise $L^2=a^2+b^2$. By Lagrange, $2a=\frac{\mu }{a^2},2b=\frac{2\mu }{b^2}\Rightarrow b^3=2a^3\Rightarrow b=2^{1/3}a$ $\begin{array}{rlr}& \frac{1}{a}+\frac{2}{2^{1/3}a}=1\Rightarrow a=1+2^{2/3},b=2^{1/3}+2=2^{1/3}\left(1+2^{2/3}\right)& L^2={\left(1+2^{2/3}\right)}^2\left(1+2^{2/3}\right)={\left(1+2^{2/3}\right)}^3\Rightarrow L={\left(2^{2/3}+1\right)}^{3/2}\end{array}$