Problem source

The line $L$ has equation $y=c-mx$, with $m>0$ and $c>0$. 
It passes through the point $R(a,b)$ and cuts the axes at the points $P(p,0)$ and $Q(0,q)$, where $a$, $b$, $p$ and $q$ are all positive. Find $p$ and $q$ in terms of $a$, $b$ and $m$. As $L$ varies with $R$ remaining fixed, show that the minimum value of the sum of the distances of $P$ and $Q$ from the origin is $(a^{\frac12} + b^{\frac12})^2$, and find in a similar form the minimum distance between $P$ and $Q$. (You may assume that any stationary values   of these distances are minima.)

Solution source

\begin{align*}
&& b &= c - ma \\
\Rightarrow && c &= b+ma \\
\Rightarrow && y &= m(a-x)+b \\
\Rightarrow && q &= ma+b \\
&& p &= \frac{ma+b}{m} \\
\\
&& d &= p+q \\
&&&= a + \frac{b}{m} + ma + b \\
\Rightarrow && d' &= -bm^{-2}+a \\
\Rightarrow && m &= \sqrt{b/a} \\
\\
\Rightarrow &&d &= a  + \sqrt{ba}+\sqrt{ba} + b \\
&&&= (\sqrt{a}+\sqrt{b})^2 \\
\\
&& |PQ|^2 &= p^2 + q^2 \\
&&&= a^2 + \frac{2ab}{m} + \frac{b^2}{m^2} + m^2a^2 + 2mab + b^2 \\
&&&= a^2+b^2 + \frac{b^2}{m^2} + \frac{2ab}{m}+ 2abm + a^2m^2 \\
&& \frac{\d}{\d m}&= -2b^2m^{-3}-2abm^{-2}+2ab + 2a^2m \\
&& 0 &=2a^2m^4+2abm^3-2abm-2b^2 \\
&&&= 2(am^3-b)(am+b) \\
\Rightarrow && m &= \sqrt[3]{\frac{b}{a}} \\
\\
&&|PQ|^2 &= \left[ a^{1/3}(a^{2/3} + b^{2/3}) \right]^2 + \left[ b^{1/3}(a^{2/3} + b^{2/3}) \right]^2 \\
&&&= a^{2/3}(a^{2/3} + b^{2/3})^2 + b^{2/3}(a^{2/3} + b^{2/3})^2 \\
&&&= (a^{2/3} + b^{2/3})^2 \cdot (a^{2/3} + b^{2/3}) \\
&&&= (a^{2/3} + b^{2/3})^3 \\
\Rightarrow && |PQ| &= (a^{2/3} + b^{2/3})^{3/2}
\end{align*}


We can also do this with AM-GM instead:

\begin{align*}
&& d &= a + b + \frac{b}{m} + am \\
&&&\geq a+b + 2 \sqrt{\frac{b}{m} \cdot am} \\
&&&= a+2\sqrt{ab}+b \\
\\
&& |PQ|^2 &= a^2+b^2 + \frac{b^2}{m^2} + \frac{2ab}{m}+ 2abm + a^2m^2 \\
&&&= a^2+b^2 + \frac{b^2}{m} + abm + abm + a^2m^2 + \frac{ab}{m} + \frac{ab}{m} \\
&&&= a^2+b^2 + 3\sqrt[3]{ \frac{b^2}{m} \cdot abm \cdot abm} + 3 \sqrt[3]{ a^2m^2 \cdot \frac{ab}{m} \cdot \frac{ab}{m} } \\
&&&= a^2 + 3b^{4/3}a^{2/3}+3b^{2/3}a^{4/3}+b^2 \\
&&&= (a^{2/3}+b^{2/3})^3
\end{align*}