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Question

P, Q, R are the points of intersection of a line / with the sides BC, CA, AB of aΔ ABC respectively, then BPPC.CQQA.ARRBis

A
1
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B
-1
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C
2
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D
-2
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Solution

The correct option is A -1
Let A(x1,y1), B(x2,y2) and C(x3,y3) be the vertices of the Δ ABC. Let the equation of the line / be ax + by + c = 0.
Suppose line / divides BC at P in the ratio m: 1 ie, BPPC=m1 Therefore, coordinates of P are (mx3+x2m+1,my3+y2m+1)
Since, P lies on ax + by + c = 0, therefore a (mx3+x2m+1)+b(my3+y2m+1)+c=0
m(ax3+by3+c)+(ax2+by2+c)=0
m1=(ax2+by2+cax3+by3+c)
BPPC=(ax2+by2+cax3+by3+c)...........(i)
SimilarlyCQQA=(ax3+by3+cax1+by1+c)...........(ii)
AndARRB=(axa+by1+cax2+by2+c)...........(iii)
Multiplying Eqs. (i), (ii) amd (iii), we get
BPPC.CQQA.ARRB=1

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