The equation of the plane through point (1, 2, -3) which is parallel to the plane 3x- 5y + 2z = 11 is given by

The correct option is: 3x – 5y + 2z + 13 = 0

Step-by-step Explanation:

1. Understanding Parallel Planes:
Two planes are parallel if they have the same normal vector. The general equation of a plane is given by:
$Ax+By+Cz+D=0$
where the vector $(A,B,C)$ represents the normal vector to the plane. Therefore, any plane parallel to the given plane:
$3x-5y+2z=11$
will have the same coefficients for x, y, and z, but a different constant term. Thus, we can write the equation of the parallel plane as:
$3x-5y+2z+d=0$
where d is a constant to be determined.

2. Finding the Constant d:
We are given that the required plane passes through the point $(1,2,-3)$. This means the coordinates $x=1$, $y=2$, and $z=-3$ must satisfy the equation of the plane. Substituting these values into our equation:

$$3\left(1\right)-5\left(2\right)+2(-3)+d=0$$

Simplify the terms:
$3-10-6+d=0$
$-13+d=0$
$d=13$

3. Writing the Final Equation:
Now, substitute the value of $d=13$ back into the equation of the parallel plane:
$3x-5y+2z+13=0$
This perfectly matches the correct option.