Question

1. find the shortest distance from the origin to the line defined by y = 3x - 10.

Explanation:

Step1: Recall the distance formula from a point to a line

The formula for the distance \( d \) from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is \( d=\frac{\vert Ax_0 + By_0 + C\vert}{\sqrt{A^2 + B^2}} \). First, we need to rewrite the given line \( y = 3x - 10 \) in the general form \( Ax+By + C = 0 \). Rearranging, we get \( 3x - y - 10 = 0 \), so \( A = 3 \), \( B=- 1 \), \( C=-10 \). The origin is \( (0,0) \), so \( x_0 = 0 \), \( y_0 = 0 \).

Step2: Substitute into the distance formula

Substitute \( A = 3 \), \( B=-1 \), \( C = - 10 \), \( x_0 = 0 \), \( y_0 = 0 \) into the formula:
\( d=\frac{\vert3\times0+(- 1)\times0+(-10)\vert}{\sqrt{3^{2}+(-1)^{2}}}=\frac{\vert - 10\vert}{\sqrt{9 + 1}}=\frac{10}{\sqrt{10}}=\sqrt{10} \) (rationalizing the denominator: \( \frac{10}{\sqrt{10}}=\frac{10\sqrt{10}}{10}=\sqrt{10} \))

Answer:

\( \sqrt{10} \)