Answer:
The distance is approximately 3.6
Step-by-step explanation:
The distance from a point (xo, yo) to a line ax+by+c=0 is the shortest distance from the given point to any point on the line.
It can be calculated with the formula:
[tex]\displaystyle d={\frac {\mid ax_{0}+by_{0}+c\mid}{\sqrt {a^{2}+b^{2}}}}[/tex]
The coordinates of point A are (1,-2). The equation of the line must be found by knowing two clear points it passes through.
These points are (0,3) and (6,-1). First, we calculate the slope:
[tex]\displaystyle m=\frac{y_2-y_1}{x_2-x_1}[/tex]
[tex]\displaystyle m=\frac{-1-3}{6-0}=\frac{-4}{6}=-\frac{2}{3}[/tex]
The equation of the line in slope-point form is:
[tex]y=m(x-x_o)+y_o[/tex]
Taking the point (0,3) and the slope above:
[tex]\displaystyle y=-\frac{2}{3}(x-0)+3[/tex]
Multiply by 3:
[tex]3y=-2x+9[/tex]
Moving all terms to the left side:
[tex]2x+3y-9=0[/tex]
We now have all the required values to calculate the distance:
a=2, b=3, c=-9, xo=1, yo=-2
[tex]\displaystyle d={\frac {\mid 2\cdot 1+3\cdot (-2)+(-9)\mid}{\sqrt {2^{2}+3^{2}}}}[/tex]
[tex]\displaystyle d={\frac {\mid 2-6-9\mid}{\sqrt {13}}}\approx 3.61[/tex]
The distance is approximately 3.6
