Question

on a golf course, three holes X(-6,-1),B and C(9;-4) lies on a straight line in that order The distance between B and C is 2 times that between B and A. Rahul strikes the Ball which is at point P (2, 3) such that it goes in the hole B. "find the cordinates of B. find the shortest distance by the ball​

Answer 1

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).