How to find P in terms of k? (image of full question below)

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1 Answer
Jul 11, 2018

P divides AB in the ratio 17/16:1=17:16
Coordinates of P ((5k-1)/(k+1),(7-2k)/(k+1))

Explanation:

A(-1,7)
B(5,-2)
P divides AB in the ratio k:1

Let P have the coordinates (x,y)

x=(-1times1+5timesk)/(k+1)

x=(5k-1)/(k+1)

y=(7times1+ktimes-2)/(k+1)

y=(7-2k)/(k+1)

Therefore, P has the coordinates ((5k-1)/(k+1),(7-2k)/(k+1))

If P lies on the line 5x-4y-1=0, then subbing in P's coordinates can help us solve for k

5times(5k-1)/(k+1)-4times(7-2k)/(k+1)-1=0

(25k-5)/(k+1)-(28-8k)/(k+1)-1=0

(25k-5)-(28-8k)-(k+1)=0

25k-5-28+8k-k-1=0

32k-34=0

32k=34

k=34/32

k=17/16

Therefore, P divides AB in the ratio 17/16:1=17:16