Point P divides the line segment joining the points X(3, 1) and Y(6, –8) such that XY/YP=4/1
If P lies on the
line 2x + y + k = 0. Find the value of k.
Answers
To divide a line segment AB in the ratio m:n, a ray AX making an acute ∠BAX is drawn and then (m+n) points are marked at equal distances on ray AX
m=3 and n=5
∴Minimum number of points to be marked on AX
⇒m+n=3+5=8
Step-by-step explanation:
it is not known whether point P divides the line segment internally or externally. first we will assume it to be internally
1. p divides internally
P is between X and Y
given XY/YP = 4/1
= (XP+YP)/YP = 4/1
XP/YP + 1 = 4
XP/YP = 3/1
the cordinate of P
=( (6*3+3*1)/(3+1), (3*-8+1*1)/(3+1))
= (21/4, -23/4)
now this point lies on 2x+y+k = 0
2*21/4+(-23/4)+k= 0
k = 23/4 - 42/4
k = -19/4
2. p divides externally
Y is between X and P
given XY/YP = 4/1
adding 1 both side
XY/YP +1 = 5
XY+YP/YP = 5
XP/YP = 5/1
coordinate of P
= ((5*6-1*3)/(5-1), (-8*5-1*1)/(5-1))
= (27/4,-41/4)
this point lies on 2x+y+k = 0
2*27/4+(-41/4)+k = 0
k = 41/4-54/4
= -13/4