   Chapter 11.5, Problem 80E

Chapter
Section
Textbook Problem

# Intersection of a Plane and a LineIn Exercises 83–86, find the point(s) of intersection (if any) of the plane and the line. Also, determine whether the line lies in the plane. 5 x + 3 y = 17 , x − 4 2 = y + 1 − 3 = z + 2 5

To determine

To calculate: Find the points of intersection of plane 5x+3y=17 and line x42=y+13=z+25 and find whether the provided line lies in the provided plane.

Explanation

Given:

The equation of the plane is,

5x+3y=17

And, the symmetric equations of the line are:

x42=y+13=z+25

Formula used:

Symmetric equations of a line are:

xx1a=yy1b=zz1c

Calculation:

First step is to convert the symmetric equation of line into the parametric equation of the line.

This can be done by equating it to the parameter 't'.

So, the parametric equations of the line are,

x42=y+13=z+25=t

The first parametric equation is,

x42=tx4=2tx=4+2t

The second parametric equation is,

y+13=ty+1=3ty=13t

The third parametric equation is,

z+25=tz+2=5tz=2+5t

So, from above calculation the parametric equations are,

x=4+2t,y=13t and z=5t2

Let us now, substitute the obtained values of x, y and z in the provided equation of plane to obtain the value of the parameter t

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