Question

If $ax+4y+3=0,bx+5y+3=0$ and $cx+6y+3=0$ are concurrent lines, then $a+c=$

• A. $3b$
• B. $2b$
• C. $b$
• D. $4b$

Answer 1

Answer: (B)

$2b$

Three straight lines $a_{1}x+b_{1}y+c_1=0$, $a_{2}x+b_{2}y+c_2=0$

and $a_{3}x+b_{3}y+c_3=0$ are concurrent if

$\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|=0$

$\left|\begin{array}{ccc}a& 4& 3\\ b& 5& 3\\ c& 6& 3\end{array}\right|=0$

$3\left|\begin{array}{ccc}a& 4& 1\\ b& 5& 1\\ c& 6& 1\end{array}\right|=0$

$\left|\begin{array}{ccc}a& 4& 1\\ b& 5& 1\\ c& 6& 1\end{array}\right|=0$

$a(5-6)-4(b-c)+1(6b-5c)=0$

$-a-4b+4c+6b-5c=0$

$-a+2b-c=0$

$a+c=2b$