1. Find k so that the point P(-4,6) lies on the line segment joining A (k,0) and B (3, -8). Also find the ratio in which P divides AB.
Answers
Step-by-step explanation:
Using the section formula, if a point (x,y) divides the line joining the points (x
1
,y
1
) and (x
2
,y
2
) in the ratio m:n, then
(x,y)=(
m+n
mx
2
+nx
1
,
m+n
my
2
+ny
1
)
Let A(1,-2) and B(-3,4) be the given points. Let the points of trisection be P and Q. Then, AP=PQ=QB=λ(say)
.∴PB=PQ+QB=2λandAQ=AP+PQ=2λ
⇒AP:PB=λ:2λ=1:2andAQ:QB=2λ:λ=2:1
So,P divides AB internally in the ratio 1:2 while Q divides internally in the ratio 2:1. Thus, the coordinates of P and Q are
P(
1+2
1×−3+2×1
,
1+2
1×4+2×−2
)=P(
3
−1
,0)
Q(
2+1
2×−3+1×1
,
2+1
2×4+1×(−2)
)=Q(
3
−5
,2)respectively
Hence, the two points of trisection are (-1/3,0) and (-5/3,2).
REMARK: As Q is the mid-point of BP. So, the coordinates of Q can also be obtained by using mid-point formula.