At what point does the line intersect the plane containing the point (0, -1, 0) which is parallel to the xz- plane?

Question

At what point does the line  intersect the plane containing the point (0, -1, 0) which is parallel to the xz- plane?

Quizplus · Accepted Answer

To find the intersection point, we need to set the $y$-coordinate of the line equal to $-1$, since the plane we are intersecting with has a $y$-coordinate of $-1$. We can solve for $t$ in the parametric equation of the line:

$$-1=8t-1 \Rightarrow t=0$$

Thus, the point of intersection occurs when $t=0$. We can substitute this into the parametric equation of the line to find the coordinate:

$$(8,1,-2)+0\cdot(9,-4,7)=(8,1,-2)$$

Now we check if this point lies in the plane. Since the plane is parallel to the xz-plane, we know that its equation has the form $y=-1$. Thus, the $y$-coordinate of the point we found must be $-1$. Since it is not, we know that this point does not lie in the plane.

To get an intersection point, we need to project the point we found onto the plane. This can be done by subtracting the distance between the point and the plane in the y-direction:

$$d=|-1-1|=2$$

Thus, the projection of the point onto the plane is:
$$(8,1-2,-2)=(8,-1,-2)$$

And this is answer choice $\boxed{	extbf{(B)}}$.

At What Point Does the Line Intersect the Plane