TITLE: point on a line closest to another point
QUESTION [1 upvotes]: Question: 
Find the point on the line $2x+y+3=0$ that is closest to the point $(2,-6)$
This question really made me think for a while as to how to approach it. 
I calculated distance as $\frac{1}{\sqrt{14}}$ using $D = \frac{ax_0+by_0+C}{\sqrt{a^2+b^2+c^2}}$
So would that mean that $x$ would be closest when the distance is $2- \frac{1}{\sqrt{14}} = 1.732738758$ 
Any suggestions?

REPLY [1 votes]: The equation for the line perpendicular to $y = -2x - 3$ passing through $(2, -6)$ is $y = 0.5x - 7$. 
($0.5$ is the negative reciprocal of $-2$ and the $y$ intercept is $-6$ minus $2\cdot 0.5$)
Subtracting one from the other, their intersection is where $0 = 2.5x - 4$, which is at $x = 1.6$ and therefore $y = -6.2$.
$d$ the distance between their intersection $(1.6, -6.2)$ and $(2, -6)$ is
$d = \sqrt{0.4^2 + 0.2^2} =  \sqrt{0.2}$
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