Answer:
The point P is (-15,0).
Step-by-step explanation:
We need to find the slope of the line L to get the coordinates of P.
We know that L is tangent to the circle at point (-3, 6), which means L is perpendicular to the radius of the circle (a line between the origin (0,0) and the tangent point (-3,6))
So let's find the slope or the radius.
[tex]m_{R}=\frac{6-0}{-3-0}[/tex]
[tex]m_{R}=-2[/tex]
Now, we know that if two lines are perpendiculars their slopes follow the rule:
[tex]m_{L}=-\frac{1}{m_{R}}[/tex]
[tex]m_{L}=\frac{1}{2}[/tex]
Knowing the slope of L we can find the equation of this line. The slope-intercept equation of a line is:
[tex]y=mx+b[/tex]
[tex]y=\frac{1}{2}x+b[/tex]
Using the point (-3,6) we can find b.
[tex]6=\frac{1}{2}(-3)+b[/tex]
[tex]b=\frac{15}{2}[/tex]
The line equation will be:
[tex]y=\frac{1}{2}x+\frac{15}{2}[/tex]
The point P has only an x-coordinate value (x,0)
So we have:
[tex]0=\frac{1}{2}x+\frac{15}{2}[/tex]
[tex]x=-15[/tex]
Therefore, the point P is (-15,0).
I hope it helps you!
