Question

The straight lines $x+2y-9=0,3x+5y-5=0$ and $ax+by-1=0$ are concurrent, if the straight line $35x-22y+1=0$ passes through the point

• A. $(a,b)$
• B. $(b,a)$
• C. $(-a,-b)$
• D. None of these

Answer 1

The correct option is A

$(a,b)$

Explanation for the correct option:

Given equations of lines are:

$x+2y-9=0...\left(i\right)$

$3x+5y-5=0...\left(ii\right)$

$ax+by-1=0...\left(iii\right)$

Since these are concurrent, by the condition for concurrency of three lines in the determinant form,

$$\begin{gathered} \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{\left|}\begin{array}{ccc}{\mathbf{a}}_{\mathbf{1}}& {\mathbf{b}}_{\mathbf{1}}& {\mathbf{c}}_{\mathbf{1}}\\ {\mathbf{a}}_{\mathbf{2}}& {\mathbf{b}}_{\mathbf{2}}& {\mathbf{c}}_{\mathbf{2}}\\ {\mathbf{a}}_{\mathbf{3}}& {\mathbf{b}}_{\mathbf{3}}& {\mathbf{c}}_{\mathbf{3}}\end{array}\mathbf{\right|}\mathbf{=}\mathbf{0} \\ \Rightarrow \left|\begin{array}{ccc}1& 2& -9\\ 3& 5& -5\\ a& b& -1\end{array}\right|=0 \\ \Rightarrow 1(-5+5b)-2(-3+5a)-9(3b-5a)=0 \\ \Rightarrow -5+5b+6-10a-27b+45a=0 \\ \Rightarrow 35a–22b+1=0...\left(iv\right) \end{gathered}$$

Given $35x-22y+1=0...\left(v\right)$

Comparing $\left(iv\right)$ with $\left(v\right)$,

We can say that $35x-22y+1=0$ passes through $(a,b).$

Hence, the correct option is option A.