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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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Start Solution

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 : \(\require{bbox} \bbox[2pt,border:1px solid black]{{\left( {\frac{{351}}{{11}}, - 7,0} \right)}}\) .