Straight Line and Pair of Straight Lines 3 Question 7
8. Three lines $p x+q y+r=0, q x+r y+p=0$ and $r x+p y+q=0$ are concurrent, if
(1985, 2M)
(a) $p+q+r=0$
(c) $p^{3}+q^{3}+r^{3}=3 p q r$
(b) $p^{2}+q^{2}+r^{2}=p r+r q$
(d) None of these
Match the Columns
Match the conditions/expressions in Column I with statement in Column II.
Show Answer
Answer:
Correct Answer: 8. $(a, c)$
Solution:
- Given lines $p x+q y+r=0, q x+r y+p=0$ and $\quad r x+p y+q=0$ are concurrent.
$\therefore \quad\left|\begin{array}{ccc}p & q & r \ q & r & p \ r & p & q\end{array}\right|=0$
Applying $R _1 \rightarrow R _1+R _2+R _3$ and taking common from $R _1$
$$ \begin{array}{rr} & (p+q+r)\left|\begin{array}{lll} 1 & 1 & 1 \\ q & r & p \\ r & p & q \end{array}\right|=0 \\ \Rightarrow & (p+q+r)\left(p^{2}+q^{2}+r^{2}-p q-q r-p r\right)=0 \\ \Rightarrow & p^{3}+q^{3}+r^{3}-3 p q r=0 \end{array} $$
Therefore, (a) and (c) are the answers.
