At what point does the line intersect the plane containing the point (-1, 0, 0) which is parallel to the xy-plane?
A) (-1, 8, 17)
B) (-1, 0,1)
C) (0, -6, 17)
D) (-1, -6, 17)
E) (2, 8, 1)

Question

At what point does the line  intersect the plane containing the point (-1, 0, 0) which is parallel to the xy-plane?
A) (-1, 8, 17) 
B) (-1, 0,1) 
C) (0, -6, 17) 
D) (-1, -6, 17) 
E) (2, 8, 1)

Quizplus · Accepted Answer

First, we can determine the direction vector of the line from the given coordinates: Direction Vector = <8, 21, 10> Next, we can find the equation of the plane containing (-1,0,0) and parallel to the xy-plane. Since the plane is parallel to the xy-plane, its normal vector will be <0,0,1> (perpendicular to the xy-plane). We can use the point-normal form of the equation of a plane: 0(x+1) + 0(y-0) + 1(z-0) = 0 → z = 0-x So the equation of the plane is z = -x. Now we need to find where the line intersects this plane. All points on the line can be represented by the parametric equations: x = 11e+0.8t y = 21e+2.1t z = 10e+t We can substitute z = -x to eliminate t: 10e+t = -11e-0.8t → 10.8t = -21e-11e → t = -(21e)/10.8 Substitute t into the parametric equations to get the point of intersection: x = 11e - 0.8(21e)/10.8 = -e y = 21e + 2.1(21e)/10.8 = 3e z = 10e - (21e)/10.8 = (87e)/54 The point (-e, 3e, (87e)/54) is on the line and on the plane, so we can substitute e = 1 to get the actual point of intersection: (-1, 3, 87/54) Out of the answer choices, only D has the correct x and z coordinates, so the answer is D.

At What Point Does the Line Intersect the Plane