Question

If the points (p,q), (m,n) and (p-m,q-n) are collinear, show that pn=qm.

Answer 1

❤♣️HËŁŁØ MÃŤÉ♣️❤

Let A(p,q) B(m,n) and C(p-m,q-n), then,

If the points are collinear, than, area of tri. ABC = 0

so, 1/2{p(n-(q-n) +m(q-n-q) + (p-m)(q-n)}

= 1/2{p(2n-q) -mn +pq -pn -mq +mn

0 = 2pn -pq -mn +pq -pn -qm +mn

so, qm = 2pn - pn

so, qm = pn.....proved..

Answer 2

SOLUTION

Here, the given Points are collinear, so

p × m × p-m × p

q × n × q-n × q

Since,

=)[p×n + m(q-n)+(p-m)q]-[m×q+(p-m)n+p(q-n)]=0

=)(pn+qm-mn+pq-mq)-(mq+pn-mn+pq-pn)=0

=)(pn+pq-mn)- (mq-mn+pq)=0

=) pn-mq=0

=) pn= qm [proved]

hope it helps ✔️