Question

Find the shortest distance from the origin to the line defined by y=3x-10

Answer 1

To find the shortest distance from the origin (0, 0) to the line defined by the equation \( y = 3x - 10 \), we can use the formula for the distance \( d \) from a point \( (x_0, y_0) \) to the line defined by \( Ax + By + C = 0 \):

\[
d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}
\]

First, we need to rewrite the line equation \( y = 3x - 10 \) in the standard form \( Ax + By + C = 0 \):

\[
3x - y - 10 = 0
\]

From this, we can identify the coefficients:

- \( A = 3 \)
- \( B = -1 \)
- \( C = -10 \)

Now we can plug in the coordinates of the origin \( (x_0, y_0) = (0, 0) \) into the distance formula:

\[
d = \frac{|3(0) + (-1)(0) - 10|}{\sqrt{3^2 + (-1)^2}} = \frac{|-10|}{\sqrt{9 + 1}} = \frac{10}{\sqrt{10}} = \frac{10}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}} = \frac{10\sqrt{10}}{10} = \sqrt{10}
\]

Thus, the shortest distance from the origin to the line \( y = 3x - 10 \) is

\[
\boxed{\sqrt{10}}
\]