In a triangle ABC, D divides BC in the ratio 3 : 2 and E divides CA in the ratio 1 : 3. The lines AD and BE meet at H and CH meets AB in F. Find the ratio in which F divides AB.
Answers
Answered by
0
ANSWER
Take A as origin and the position vectors of B and C be b and c.
Hence the position vectors of other points under given conditions are
D=
5
3c+2b
,E=
4
1.0+3c
=
4
3
c.
Equations of AD and BE are
AD is r=t
5
3c+2b
BE is r=(1−s)b+s⋅
4
3
c.
They intersect at H.
Comparing coefficients of b and c, we get
5
2
t=1−s,
5
3
t=
4
3
s.
∴s=
5
4
t.
∴
5
2
t+
5
4
t=1
∴t=
6
5
,s=
6
4
Point H is
6
3c+2b
.
Now F is point of inersection of AB and CH whose equations are r=tb
and r=(1−s)c+s
6
3c+2b
Comparing the coefficients,
t=
6
2s
=
3
s
and 1−s+
6
3
s=0⇒s=2⇒t=
3
1
s=
3
2
∴ P.V. of F=
3
2
b or
AF
=
3
2
b,
FB
=b−
3
2
b=
3
1
b
∴AF:FB=2:1
I hope it will help you
Mark me as brainliest
Similar questions