Question

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.

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Answer 1

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

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