Question

Which of the following lines is concurrent with the lines $3\mathrm{x}+4\mathrm{y}+6=0$ and $6\mathrm{x}+5\mathrm{y}+9=0$?

• A. $2\mathrm{x}+3\mathrm{y}+5=0$
• B. $3\mathrm{x}+3\mathrm{y}+5=0$
• C. $7\mathrm{x}+9\mathrm{y}+3=0$
• D. None of these

Answer 1

The correct option is B

$3\mathrm{x}+3\mathrm{y}+5=0$

Explanation for the correct option: b) $3\mathrm{x}+3\mathrm{y}+5=0$.

Step 1:Find the point of intersection of $3\mathrm{x}+4\mathrm{y}+6=0$ and $6\mathrm{x}+5\mathrm{y}+9=0$.

The lines which intersect each other at exactly one point are the concurrent lines.

$$\begin{gathered} 3\mathrm{x}+4\mathrm{y}=-6\left(\mathrm{i}\right) \\ 6\mathrm{x}+5\mathrm{y}=-9\left(\mathrm{ii}\right) \end{gathered}$$

Multiply $\left(\mathrm{i}\right)$ by $2$.

$6\mathrm{x}+8\mathrm{y}=-12\left(\mathrm{i}\right)$

$6\mathrm{x}+5\mathrm{y}=-9\left(\mathrm{ii}\right)$

Subtract $\left(\mathrm{ii}\right)$ from $\left(\mathrm{i}\right)$

$\Rightarrow$ $3\mathrm{y}=-3$

$\Rightarrow$ $\mathrm{y}=-1$

Put $\mathrm{y}=-1$in $\left(\mathrm{i}\right)$.

$3\mathrm{x}+4\left(-1\right)=-6$

$\Rightarrow$ $3\mathrm{x}-4=-6$

$\Rightarrow$ $3\mathrm{x}=-2$

$\Rightarrow$ $\mathrm{x}=-\frac{2}{3}$

The point of intersection is $\left(-\frac{2}{3},-1\right)$.

Step 2: Put the values of $\mathrm{x}$ and $\mathrm{y}$ in $3\mathrm{x}+3\mathrm{y}+5=0$.

$3\mathrm{x}+3\mathrm{y}+5=0$

$\Rightarrow$$3\left(-\frac{2}{3}\right)+3\left(-1\right)+5=0$

$\Rightarrow$ $-2-3+5=0$.

Since the left-hand side is equal to the right-hand side, it is concurrent.

Hence, the correct answer is option b) $3\mathrm{x}+3\mathrm{y}+5=0$.

Explanation of incorrect options:

Option (A)

Putting $\mathrm{x}=-\frac{2}{3}$ and $\mathrm{y}=-1$ in the equation $2\mathrm{x}+3\mathrm{y}+5=0$

$2\left(-\frac{2}{3}\right)+3\left(-1\right)+5=0$

$\Rightarrow$ $-\frac{4}{3}-3+5=0$

$\Rightarrow$ $\frac{2}{3}=0$

Which is incorrect.

Option (C)

Putting $\mathrm{x}=-\frac{2}{3}$ and $\mathrm{y}=-1$ in the equation $7\mathrm{x}+9\mathrm{y}+3=0$

$7\left(-\frac{2}{3}\right)+9\left(-1\right)+3=0$

$\Rightarrow$ $-\frac{14}{3}-9+5=0$

$\Rightarrow$ $-\frac{26}{3}=0$

Which is incorrect.

Hence, the correct answer is option b) $3\mathrm{x}+3\mathrm{y}+5=0$.