Question

If the lines $x+py+p=0,qx+y+q=0$ and $rx+ry+1=0(p,q,r$ being distinct and $\ne$ 1) are concurrent, then the value of
$\frac{p}{p-1}+\frac{q}{q-1}+\frac{r}{r-1}=$

• A. $1$
• B. $-1$
• C. $2$
• D. $-2$

Answer 1

Answer: (A)

$1$

$x+py+p=0$

$qx+y+q=0$

$rx+ry+1=0$

$\left|\begin{array}{ccc}1& p& p\\ q& 1& q\\ r& r& 1\end{array}\right|$

$(1-qr)-p(q-pqr)+p(qr-r)=0$

$1-qr-pq+pqr+pqr-pr=0$

$qr(p-1)+pr(q-1)-pq(r-1)-pqr=0$

$\Rightarrow \frac{p}{p-1}+\frac{q}{q-1}+\frac{r}{r-1}=1$

Option A