Question

Find the shortest distance from a hotel located at $\left(2,4\right)$ to a straight road with equation $3x+5y+8=0$.

Answer 1

Calculation of shortest distance:

The shortest distance between the points $\left(x_0,y_0\right)$ and the line $ax+by+c=0$ is $\left|\frac{a{x}_0+b{y}_0+c}{\sqrt{a^2+b^2}}\right|$.

Calculate the shortest distance between the point $\left(2,4\right)$ and $3x+5y+8=0$.

$$\begin{gathered} \left|\frac{3\times 2+5\times 4+8}{\sqrt{3^2+5^2}}\right|=\left|\frac{6+20+8}{\sqrt{9+25}}\right| \\ \Rightarrow =\left|\frac{34}{\sqrt{34}}\right| \\ \therefore =\sqrt{34} \end{gathered}$$

Hence, shortest distance is $\sqrt{\mathbf{34}}$.