I encountered this problem:
A straight line passes through a fixed point $(h,k) \ (h,k>0)$ and intersects the coordinate axes at $P(a,0)$ and $Q(0,b)$. Show that the minimum length of $PQ$ is $(h^{2/3}+k^{2/3})^{3/2}$.
Now, I proceeded like this:
If a line passes through $(h,k)$ then its equation is $$\frac{y-k}{x-h}=m.$$
If it passes through $(a,0)$ and $(0,b)$ then we have $a=h-\frac{k}{m}$ and $b=k+mh$.
Length of $PQ$ is $\sqrt{a^2+b^2}$.
We can put the value of $a$ and $b$ in the expression and then differentiate w.r.t. $m$, to find out the minimum value.
But after equating the first differential to $0$, I found two roots of $m$. As the values of $h$ and $k$ can be positive or negative, I cannot put them in the second differential to find out which one is minimum and which one is maximum.
What am I doing wrong? and, Is there any other, simpler method that can be used?
Equation of line $\mathcal{AB}$ is $\displaystyle \frac{x}{a}+\frac{y}{b} =1$
Line passes through fixed point $(h,k)$
Then $\displaystyle \frac{h}{a}+\frac{k}{b} =1$
Use Holder,s Inequality
$$(a^2+b^2)\bigg(\frac{h}{a}+\frac{k}{b}\bigg)\bigg( \frac{h}{a}+\frac{k}{b}\bigg)\geq \bigg(h^{\frac{2}{3}}+k^{\frac{2}{3}}\bigg)^3$$
$$\mathcal{\min{PQ}}=\sqrt{a^2+b^2}\geq \bigg(h^{\frac{2}{3}}+k^{\frac{2}{3}}\bigg)^{\frac{3}{2}}$$