1. The coordinates of point B are (11, -22/5).
2. The shortest distance traveled by the ball is 13 sqrt(5).
Here is the solution to the problem:
Part 1: Find the coordinates of point B
Given that the distance between B and C is twice the distance between B and A, we can set up the following equation:
BC = 2AB
where BC is the distance between point B and point C, and AB is the distance between point A and point B. We can use the distance formula to calculate these distances:
BC = sqrt((9 - x_B)^2 + (-4 - y_B)^2)
AB = sqrt((x_B - (-6))^2 + (y_B - (-1))^2)
Substituting these expressions into the original equation, we get:
sqrt((9 - x_B)^2 + (-4 - y_B)^2) = 2 sqrt((x_B - (-6))^2 + (y_B - (-1))^2)
Simplifying this equation, we get:
3x_B - 2y_B - 13 = 0
This equation represents the relationship between the x-coordinate and y-coordinate of point B. We can also use the fact that point B, point A, and point C lie on a straight line to get another equation:
y_B = m(x_B) + b
where m is the slope of the line and b is the y-intercept of the line. We can find the slope of the line by using the coordinates of point A and point C:
m = (C[1] - A[1]) / (C[0] - A[0]) = (-4 - (-1)) / (9 - (-6)) = -3/15 = -1/5
We can find the y-intercept of the line by substituting the coordinates of point A into the equation for the line:
-1 = (-1/5)(-6) + b
-1 = 6/5 + b
b = -11/5
Now we have two equations that we can use to solve for the coordinates of point B. Substituting the equation for y_B into the equation for point B on a straight line, we get:
m(x_B) + b = 0
(-1/5)x_B - 11/5 = 0
x_B = 11
Now we can substitute this value of x_B back into the equation for y_B to find y_B:
y_B = m(x_B) + b
y_B = (-1/5)(11) - 11/5
y_B = -22/5
Therefore, the coordinates of point B are (11, -22/5).
Part 2: Find the shortest distance traveled by the ball
The shortest distance traveled by the ball is the straight-line distance from point P to point B. We can use the distance formula to calculate this distance:
d = sqrt((11 - 2)^2 + ((-22/5) - 3)^2)
d = sqrt(89 + 1044/25)
d = sqrt(1633/25)
d = 13/5 sqrt(25)
d = 13 sqrt(5)
Therefore, the shortest distance traveled by the ball is 13 sqrt(5).