A and C are mutually exclusive, both alternatives cannot be correct

again only A & B is the answer

Let's analyze the two lines:
1. Line L: $\(6 x+4 y+5=0\)$
2. Line M: $\(9 x+6 y=0\)$

Step 1: Check if the two lines intersect
Lines intersect if they are not parallel.
- Slope of line L :

Rewrite $\(6 x+4 y+5=0\)$ as:

$$
\(4 y=-6 x-5 \Longrightarrow y=-\frac{6}{4} x-\frac{5}{4}=-\frac{3}{2} x-\frac{5}{4}\)
$$


Slope of $L$ is $\(-\frac{3}{2}\)$.
- Slope of line M :

Rewrite $\(9 x+6 y=0\)$ as:

$$
\(6 y=-9 x \Longrightarrow y=-\frac{9}{6} x=-\frac{3}{2} x\)
$$


Slope of M is also $\(-\frac{3}{2}\)$.
Since both lines have the same slope, the lines are parallel.
Are they coincident?
Check by comparing the constants:
The general form ratio of coefficients for coincidence should be equal:
For L: $\(6 x+4 y+5=0\)$

For M : $\(9 x+6 y+0=0\)$
Compute ratios $\(\frac{6}{9}=\frac{2}{3}, \frac{4}{6}=\frac{2}{3}\)$, but constant term ratio:

$$
\(\frac{5}{0} \quad(\text { undefined })\)
$$


So lines are parallel but not coincident.
Step 2: Check if the lines intersect
Since they are parallel and not coincident, they do not intersect.
Thus, Option A is true.
Step 3: Check if line $M$ passes through origin
Equation of M: $\(9 x+6 y=0\)$
If we put $x=0, y=0$, the equation holds:

$$
\(9(0)+6(0)=0\)
$$


So the line $M$ passes through the origin.
Thus, Option B is true.
Step 4: Check if $L$ and $M$ intersect in the third quadrant
Since the lines do not intersect at all, they cannot intersect in any quadrant.
Thus, Option C is false.
Final answer:
- A. True
- B. True
- C. False

Fixed the question because the source reported ALL three answer choices are true but indeed onloy A and B