<s>[INST] Find the point $(x_0, y_0)$ on the line $ax + by = c$ that is closest to the origin. Find the point  $(x_0, y_0)$  on the line  $ax + by = c$  that is closest to the origin. 
According to this source, I thought that $\left( -\frac{ac}{a^{2}+b^{2}}, -\frac{bc}{a^{2}+b^{2}} \right)$ was the point but it doesn't seem to be correct. Thanks for any help.
 [/INST] $$(h-0)^2+\left(\dfrac{c-ah}b\right)^2=\dfrac{(a^2+b^2)h^2-2cah+c^2}{b^2}$$
Now $(a^2+b^2)h^2-2cah=\left(h\sqrt{a^2+b^2}-\dfrac{ca}{\sqrt{a^2+b^2}}\right)^2-\dfrac{(ca)^2}{a^2+b^2}\ge-\dfrac{(ca)^2}{a^2+b^2}$
the minimum value occurs iff
the equality occurs if $$h\sqrt{a^2+b^2}-\dfrac{ca}{\sqrt{a^2+b^2}}=0\iff h=?$$
Can you take it from here?
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