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### Section 12.2 : Equations of Lines

6. Does the line given by $$x = 9 + 21t$$, $$y = - 7$$, $$z = 12 - 11t$$ intersect the $$xy$$-plane? If so, give the point.

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If the line intersects the $$xy$$-plane there will be a point on the line that is also in the $$xy$$‑plane. Recall as well that any point in the $$xy$$-plane will have a $$z$$ coordinate of $$z = 0$$.

Show Step 2

So, to determine if the line intersects the $$xy$$-plane all we need to do is set the equation for the $$z$$ coordinate equal to zero and solve it for $$t$$, if that’s possible.

Doing this gives,

$12 - 11t = 0\hspace{0.25in}\hspace{0.25in} \to \hspace{0.25in}\hspace{0.25in}t = \frac{{12}}{{11}}$ Show Step 3

So, we were able to solve for $$t$$ in this case and so we can now say that the line does intersect the $$xy$$-plane.

Show Step 4

All we need to do to finish this off this problem is find the full point of intersection. We can find this simply by plugging $$t = \frac{{12}}{{11}}$$ into the $$x$$ and $$y$$ portions of the equation of the line.

Doing this gives,

$x = 9 + 21\left( {\frac{{12}}{{11}}} \right) = \frac{{351}}{{11}}\hspace{0.25in}\hspace{0.25in}y = - 7$

The point of intersection is then : $$\require{bbox} \bbox[2pt,border:1px solid black]{{\left( {\frac{{351}}{{11}}, - 7,0} \right)}}$$ .