Which of the following planes is the equation for the pane with an x-intercept at P(2,0,0), a y-intercept at Q(0,-3,0), and is parallel to the z-axis?
a) 3x-2y-6=0
b) 2x-3y+z-6=0
c) 3y-2z-6=0
d) z=-6
Only the equation without a z in it can represent a plane parallel to the z axis.
To determine which of the given plane equations satisfies the given conditions, we need to analyze the intercepts and the parallel requirement.
Let's analyze each option one by one:
a) 3x - 2y - 6 = 0:
To find the x-intercept, we set y and z to 0 and solve for x:
3x - 2(0) - 6 = 0
3x - 6 = 0
3x = 6
x = 2
So, the given plane intersects the x-axis at P(2,0,0), which is one of the given conditions. However, we also have to check if the plane is parallel to the z-axis.
To determine parallelism with the z-axis, we need to check if the equation contains the variable z. This equation does not include z, so it is not parallel to the z-axis.
b) 2x - 3y + z - 6 = 0:
Although this equation contains z and satisfies the second condition by having a variable dependent on z, we still need to check the intercepts.
To find the x-intercept, we set y and z to 0 and solve for x:
2x - 3(0) + 0 - 6 = 0
2x - 6 = 0
2x = 6
x = 3
So, this plane does not pass through P(2,0,0), violating the first condition.
c) 3y - 2z - 6 = 0:
This equation does not contain x, so it fails the first condition.
d) z = -6:
This equation represents a plane parallel to the z-axis since z has a fixed value of -6. However, we still need to check whether it satisfies the intercept condition.
To find the y-intercept, we set x and z to 0 and solve for y:
0 - 3y + (-6) - 6 = 0
-3y - 12 = 0
-3y = 12
y = -4
So, this plane passes through Q(0,-3,0) and is parallel to the z-axis. It satisfies both conditions.
Therefore, the correct answer is d) z = -6.