3. If $x$ and $y$ are real numbers, find the minimum value of $\sqrt{4+y^{2}}+\sqrt{x^{2}+y^{2}-4 x-4 y+8}+\sqrt{x^{2}-8 x+17}$.
(1 mark)
若 $x 、 y$ 為實數, 求 $\sqrt{4+y^{2}}+\sqrt{x^{2}+y^{2}-4 x-4 y+8}+\sqrt{x^{2}-8 x+17}$ 的最小值。
(1 分)
3. 5
3. Note that $x^{2}+y^{2}-4 x-4 y+8=(x-2)^{2}+(y-2)^{2}$ and $x^{2}-8 x+17=(x-4)^{2}+1$. Hence if we let $A=(0,0), B=(2, y), C=(x, 2)$ and $D=(4,3)$, the expression in the question is equal to $A B+B C+C D$. This 'total length' is minimised when $A, B, C, D$ are collinear (which is clearly possible: just take $B$ to be the intersection of the line segment $A D$ and the line $x=2$, and likewise for $C$ ), and the minimum 'total length' is the length of $A D$, which is $\sqrt{(4-0)^{2}+(3-0)^{2}}=5$.