In a triangle ABC,D and E divide the sides BC and CA in the ratio 2:1 respectively. If P is the point of intersection of AD and BE then the ratio in which P divides AD is
Let A,B,C,D,E,P have position vectors a,b,c,d, e and p respectively. Given, D and E divide the line segment BC and AE in the ratio 2:1. By section formula, ‌d=‌
2c+b
2+1
⇒3d=2c+b ‌e=‌
c+2a
2+1
⇒3e=c+2a From Eq. (i), ‌‌3d−b=2c From Eq. (ii), 3e −2a=c ∴‌‌6e−4a=2c Equating both values of 2c, 3d−b=6e−4a ⇒‌‌3d+4a=6e+b ⇒‌‌‌
3d+4a
3+4
=‌
6e+b
6+1
LHS is the position vector of the point which divides segment AD internally in the ratio 3:4. RHS is the position vector of the point which divides segment BE internally in the ratio 6:1. But P is the point of intersection of AD and BE. ∴P divides AD internally in the ratio 3:4 and P divides BE internally in the ratio 6:1.