Question

Point P divides the line segment joining the points A (2,1) and B (5, -8) such that AP : AB = 1:3. If P lies on the line 2x - y + k = 0, then find the value of k.

a) k = -3
b) k = -5
c) k = 3
d) k = 5

Answer 1

The value of k is -3.25.

Step-by-step explanation:

To find the value of k, we need to find the coordinates of point P using the given ratio. First, we calculate the difference in x-coordinates and y-coordinates between points A and B:

xB - xA = 5 - 2 = 3

yB - yA = -8 - 1 = -9

Next, we use the ratio of AP : AB = 1:3 to find the coordinates of point P:

xP = xA + (1/4)(xB - xA) = 2 + (1/4)(3) = 2 + 3/4 = 2.75

yP = yA + (1/4)(yB - yA) = 1 + (1/4)(-9) = 1 - 9/4 = -1.25

So, point P has the coordinates (2.75, -1.25). Since point P lies on the line 2x - y + k = 0, we substitute the x and y values of P into the equation:

2(2.75) - (-1.25) + k = 0

Simplifying, we get k = -3.25.