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A(x1,y1), B(x2,y2),C(x3,y3) are three ve...

`A(x_1,y_1), B(x_2,y_2),C(x_3,y_3)` are three vertices of a triangle ABC, `lx+my+n=0` is an equation of line L. If L intersects the sides BC,CA and AB of a triangle ABC at P,Q,R respectively, then `(BP)/(PC)xx(CQ)/(QA)xx(AR)/(RB)` is equal to

A

`-1`

B

`-(1)/(2)`

C

`(1)/(2)`

D

1

Text Solution

Verified by Experts

The correct Answer is:
A
Let `BP: PC = alpha: 1`
`:. P ((alpha x_(3)+x_(2))/(alpha+1),(alphay_(3)+y_(2))/(alpha+1))`
P lies on `lx +my +n = 0`
`:. L ((alpha x_(3)+x_(2))/(alpha+1)) +m ((alphay_(3)+y_(2))/(alpha+1)) +n = 0`
`:. alpha =(-(lx_(2)+my_(2)+n))/((lx_(3)+my_(3)+n))`
Similarly, if `CQ: QA = beta:1`
`:. beta = (-(lx_(3)+my_(3)+n))/((lx_(1)+my_(1)+n))`
and if `AR: RB = gamma:1`
`:. gamma =(-(lx_(1)+my_(1)+n))/((lx_(2)+my_(2)+n))`
`:. (BP)/(PC).(CQ)/(QA).(AR)/(RB) =-1`
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Knowledge Check

  • A(x_(1),y_(1)), B(x_(2),y_(2)), C(x_(3),y_(3)) are three vertices of a triangle ABC. lx +my +n = 0 is an equation of the line L. If P divides BC in the ratio 2:1 and Q divides CA in the ratio 1:3 then R divides AB in the ratio (P,Q,R are the points as in problem 1)

    A
    2:3 internally
    B
    2:3 externally
    C
    3:2 internally
    D
    3:2 externally

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