$\text{Find equation of a line through the point }(-2,1,3)\text{and parallel to the line}\frac{x-2}{4}=\frac{y+3}{-3},z=-2$

The correct option is:
$\frac{x+2}{4}=\frac{y-1}{-3},z=3$

Step-by-Step Explanation:

1. Identify the given line and its direction:
The given line has the equations:
$\frac{x-2}{4}=\frac{y+3}{-3},z=-2$
This represents a line in three-dimensional space. The equations show that:
- The x and y coordinates are related through ratios with denominators 4 and -3, which represent the direction ratios of the line in the xy-projection.
- The z-coordinate is constant at every point on the line ($z=-2$). This means the line lies in the plane parallel to the xy-plane at height $z=-2$, and its direction vector has a z-component of 0.
Thus, the direction ratios of the given line are proportional to:
$(a,b,c)=(4,-3,0)$

2. Determine the direction ratios of the parallel line:
Since the required line is parallel to the given line, it must have the same direction ratios. Therefore, the direction ratios of the required line are also:
$(a,b,c)=(4,-3,0)$

3. Formulate the equation of the line through the given point:
The line passes through the point:
$(x_1,y_1,z_1)=(-2,1,3)$
A line passing through a point $(x_1,y_1,z_1)$ with direction ratios $(a,b,c)$ is written as:
$\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}$
Substituting $a=4$, $b=-3$, and $c=0$ into the formula:
- For the x and y coordinates:
$\frac{x-(-2)}{4}=\frac{y-1}{-3}$
which simplifies to:
$\frac{x+2}{4}=\frac{y-1}{-3}$
- Since the z-component of the direction ratio is 0 ($c=0$), the z-coordinate remains constant along the entire line. Thus, the equation for z is:
$z=z_1=3$

4. Conclusion:
Combining the two parts gives the complete equation of the parallel line:
$\frac{x+2}{4}=\frac{y-1}{-3},z=3$