Question

If the points $(p,q)(m,n)$ and $(p-m,q-n)$ are collinear, Determine the relation between $p,q,n$ and $m$.

Answer 1

$\left|\begin{array}{ccc}p& q& 1\\ m& n& 1\\ p-m& q-n& 1\end{array}\right|=0$

$\Rightarrow p(n-q+n)-q(m-p+m)+1(mq-mn-np+nm)=0$

$\Rightarrow p(2n-q)-q(2m-p)+1(mq-np)=0$

$\Rightarrow 2np-pq-2qm+pq+mq-np=0$

$\Rightarrow np-qm=0$ or $np=qm$ or $\frac{p}{q}=\frac{m}{n}$ is the required relation