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 \((x_0,y_0)\) to a line \(Ax + By+ C = 0\)

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

Step2: Substitute the values into the distance formula

Substitute \(x_0 = 0\), \(y_0 = 0\), \(A = 3\), \(B=-1\), and \(C = - 10\) into the formula \(d=\frac{\vert Ax_0+By_0 + C\vert}{\sqrt{A^2 + B^2}}\). We have \(d=\frac{\vert3\times0+(-1)\times0-10\vert}{\sqrt{3^2+(-1)^2}}=\frac{\vert - 10\vert}{\sqrt{9 + 1}}=\frac{10}{\sqrt{10}}\).

Step3: Simplify the expression

Simplify \(\frac{10}{\sqrt{10}}\). We can rationalize the denominator or simplify the fraction. \(\frac{10}{\sqrt{10}}=\sqrt{10}\) (because \(\frac{10}{\sqrt{10}}=\frac{\sqrt{10}\times\sqrt{10}}{\sqrt{10}}=\sqrt{10}\)).

Answer:

\(\sqrt{10}\)