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

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Answer 1 · 2018-07-11T01:27:24

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

#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#