<s>[INST] Calculate $\sqrt{x^2+y^2+2x-4y+5} + \sqrt{x^2+y^2-6x+8y+25}$, if $3x+2y-1=0$ As the title says, given $x,y \in \mathbb{R}$ where $3x+2y-1=0$ and $x \in [-1, 3]$, calculate $A = \sqrt{x^2+y^2+2x-4y+5} + \sqrt{x^2+y^2-6x+8y+25}$. 
I tried using the given condition to reduce the complexity of the roots, but couldn't get rid of them.
 [/INST] Hint: make a geometrical interpretation and draw a picture:
$x^2+y^2+2x-4y+5=(x+1)^2 +(y-2)^2$ and consider the point $P(-1,2)$.
$x^2+y^2-6x+8y+25=(x-3)^2+(y+4)^2$ and consider the point $Q(3,-4)$.
Observe that $P$ and $Q$ are on the line $3x+2y-1=0$. 
Can you now see what $A$ means on the given interval?
The answer is here (mouse over) but try to find it yourself first:

$2\sqrt{13}$

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