How to find, the shortest distance from $O(0,0)$ to the line passing through $A(2,3)$ and $B(3,5)$.

Question

I tried to solve this problem by distance formula and I get $\sqrt5$ . But this answer is incorrect.

Answer 1

$$ d=\frac{|\vec{OA}\times\vec{OB}|}{|\vec{AB}|}=\frac{|(x_A-x_O)(y_B-y_O)-(x_B-x_O)(y_A-y_O)|}{\sqrt{(x_B-x_A)^2+(y_B-y_A)^2}}\\ =\frac{|2\cdot5-3\cdot3|}{\sqrt{(3-2)^2+(2-1)^2}}=\frac1{\sqrt5}. $$

Answer 2

The equation of $AB$ it's $$2x-y-1=0,$$ which gives that the distance it's: $$\frac{|2\cdot0-0-1|}{\sqrt{2^2+(-1)^2}}.$$ Can you end it now?

Answer 3

The line through $(2,3)$ and $(3,5)$ is $y=2x-1$, or equivalently $2x-y-1=0$

Thus the distance of $(0,0)$ from this line is

$$\frac{|2 \cdot 0 + (-1) \cdot 0 + (-1)|}{\sqrt{2^2+(-1)^2}}=\frac{1}{\sqrt{5}}=\frac{\sqrt{5}}{5} $$