Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 13 - Vector Geometry - 13.5 Planes in 3-Space - Exercises - Page 685: 53


$$3\lambda x+by+2 \lambda z=5\lambda.\\ \lambda\neq 0.$$

Work Step by Step

The equation of any plane that intersects the $ xz $-plane in the line $3x+2z=5$ is given by $$ ax+by+cz=d.$$ Now, to find the $ xz $ intersection, we put $ y=0$ and hence we get $$ ax+ cz=d.$$ Due to the intersection, we have $$ a=3\lambda, \quad c=2\lambda, \quad d=5\lambda, \quad \lambda\neq 0.$$ So the equation of any plane whose intersection is the line $3x+2z=5$ is given by $$3\lambda x+by+2 \lambda z=5\lambda.$$
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