Find the value of P if the line joining (P, 4) and (6, -2) is perpendicular to the line joining (2, P) and (-1, 3).
A. 4
B. 6
C. 3
D. O
To find the value of P, we need to find the equation of each line and then examine the slopes of the lines.
The slope of the line joining (P, 4) and (6, -2) can be found using the slope formula:
slope = (y2-y1)/(x2-x1)
slope = (-2-4)/(6-P)
slope = -6/(6-P)
The slope of the line joining (2, P) and (-1, 3) can also be found using the slope formula:
slope = (3-P)/(-1-2)
slope = (3-P)/(-3)
Since the two lines are perpendicular, the product of their slopes will be -1:
(-6/(6-P)) * ((3-P)/(-3)) = -1
Multiplying both sides by -3 and 6-P, we get:
-6(3-P) = (-1)(-3)(6-P)
-18 + 6P = 9(6-P)
-18 + 6P = 54 - 9P
Adding 9P to both sides and adding 18 to both sides, we get:
15P = 72
Dividing both sides by 15, we get:
P = 72/15
P = 4.8
Since P must be a whole number, the value of P is 5.
Answer: D. 0
To find the value of P, we first need to find the slopes of both lines.
The slope of the line joining (P, 4) and (6, -2) is given by the formula:
slope1 = (y2 - y1) / (x2 - x1)
Substituting the coordinates of the two points, we get:
slope1 = (-2 - 4) / (6 - P)
The slope of the line joining (2, P) and (-1, 3) is given by the same formula:
slope2 = (3 - P) / (-1 - 2)
Since the two lines are perpendicular to each other, their slopes are negative reciprocals of each other. Therefore, we can set up the following equation:
slope1 * slope2 = -1
((-2 - 4) / (6 - P)) * ((3 - P) / (-1 - 2)) = -1
Simplifying, we get:
((-6) / (6 - P)) * ((3 - P) / (-3)) = -1
(-6 * (3 - P)) / (6 - P) = -1
Multiplying both sides of the equation by (6 - P), we get:
-6 * (3 - P) = -(6 - P)
Expanding and simplifying, we get:
-18 + 6P = -6 + P
Rearranging terms, we get:
5P = 12
Dividing both sides by 5, we find:
P = 12/5
The value of P is not among the given answer choices. Therefore, the answer is none of the options provided.