Question

find the distance from point a(-9, -3) to the given line. y = x - 6

Explanation:

Step1: Recall the distance formula

The distance $d$ from a point $(x_0,y_0)$ to a line $Ax + By+C = 0$ is given by $d=\frac{\vert Ax_0+By_0 + C\vert}{\sqrt{A^{2}+B^{2}}}$. First, rewrite the line $y=x - 6$ in the general form $x-y-6=0$. Here, $A = 1$, $B=-1$, $C=-6$, and the point $A$ has coordinates $(x_0,y_0)=(-9,-3)$.

Step2: Substitute values into the formula

Substitute $x_0=-9$, $y_0=-3$, $A = 1$, $B=-1$, and $C=-6$ into the distance formula:
\[

$$\begin{align*} d&=\frac{\vert1\times(-9)+(-1)\times(-3)+(-6)\vert}{\sqrt{1^{2}+(-1)^{2}}}\\ &=\frac{\vert-9 + 3-6\vert}{\sqrt{1 + 1}}\\ &=\frac{\vert-12\vert}{\sqrt{2}}\\ &=\frac{12}{\sqrt{2}} \end{align*}$$

\]

Step3: Rationalize the denominator

Multiply the numerator and denominator by $\sqrt{2}$:
\[
d=\frac{12\sqrt{2}}{2}=6\sqrt{2}
\]

Answer:

$6\sqrt{2}$