Suppose the equation of P
 is x+y-z = 11 and the parametric equation of a line l is x=1+t,y=3−2t,z=2+4t. If they intersect at the point (-a, b, -c), find a, b, c

We are given the equation of the plane \( P \) as \( x + y - z = 11 \), and the parametric equations of a line \( l \) as:

\[
x = 1 + t, \quad y = 3 - 2t, \quad z = 2 + 4t
\]

To find the intersection point between the plane and the line, we substitute the parametric equations of the line into the equation of the plane. The plane equation is:

\[
x + y - z = 11
\]

Substitute \( x = 1 + t \), \( y = 3 - 2t \), and \( z = 2 + 4t \) into this equation:

\[
(1 + t) + (3 - 2t) - (2 + 4t) = 11
\]

Simplify this equation:

\[
1 + t + 3 - 2t - 2 - 4t = 11
\]
\[
(1 + 3 - 2) + (t - 2t - 4t) = 11
\]
\[
2 - 5t = 11
\]

Solve for \( t \):

\[
-5t = 11 - 2
\]
\[
-5t = 9
\]
\[
t = -\frac{9}{5}
\]

Now that we have \( t = -\frac{9}{5} \), substitute this value back into the parametric equations to find the coordinates of the intersection point:

For \( x \):
\[
x = 1 + t = 1 - \frac{9}{5} = \frac{5}{5} - \frac{9}{5} = -\frac{4}{5}
\]

For \( y \):
\[
y = 3 - 2t = 3 - 2 \times \left( -\frac{9}{5} \right) = 3 + \frac{18}{5} = \frac{15}{5} + \frac{18}{5} = \frac{33}{5}
\]

For \( z \):
\[
z = 2 + 4t = 2 + 4 \times \left( -\frac{9}{5} \right) = 2 - \frac{36}{5} = \frac{10}{5} - \frac{36}{5} = -\frac{26}{5}
\]

Therefore, the intersection point is \( \left( -\frac{4}{5}, \frac{33}{5}, -\frac{26}{5} \right) \).

We are asked to find \( a \), \( b \), and \( c \) where the point is \( (-a, b, -c) \). Thus:

\[
a = \frac{4}{5}, \quad b = \frac{33}{5}, \quad c = \frac{26}{5}
\]

Would you like further details on any step? Here are 5 related questions:

1. How do you find the parametric equations of a line given two points?
2. What is the general form of a plane equation?
3. How do you determine if two lines are parallel or intersecting?
4. Can a line and a plane be parallel? How would that affect their equations?
5. How do parametric equations relate to vector equations of lines?

**Tip:** Always double-check the signs and coefficients when solving for parameters in parametric equations, as small errors can affect the final result significantly!

Suppose the equation of P is x + y - z = 11 and the parametric equation of a line l is x = 1 + t, y = 3 - 2t, z = 2 + 4t. If they intersect at the point (-a, b, -c), find a, b, c.

Discover how to find the intersection of a line and a plane using parametric equations. Solve for the parameters and determine the coordinates of the intersection point. This detailed solution walks through the process, involving geometry and linear algebra, suitable for Grades 10-12.

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