The equation of the plane through the origin and parallel to the plane 3x – 4y + 5z + 6 = 0

The correct option is 3x – 4y + 5z = 0.

Here is the step-by-step explanation of how to find the equation of the parallel plane:

Step 1: Understand the condition for parallel planes
Two planes are parallel if their normal vectors are pointing in the same or opposite directions. This means their equations differ only by a constant term.
If the equation of a given plane is:
$$Ax+By+Cz+D=0$$ then any plane parallel to it can be represented in the general form:
$$Ax+By+Cz+k=0$$ where $k$ is a constant representing the shift along the direction of the normal vector.

Step 2: Apply this to the given equation
We are given the equation of the plane:
$$3x–4y+5z+6=0$$ Therefore, the equation of any plane parallel to this given plane will have the form:
$$3x–4y+5z+k=0$$

Step 3: Solve for the constant term using the given point
We are told that the required plane passes through the origin. The coordinates of the origin are (0, 0, 0).
By substituting $x=0$, $y=0$, and $z=0$ into our parallel plane equation, we get:
$$3\left(0\right)–4\left(0\right)+5\left(0\right)+k=0$$ Simplifying this expression gives:
$$k=0$$

Step 4: Write the final equation
Substituting $k=0$ back into the equation of the parallel plane, we obtain:
$$3x–4y+5z=0$$