The parametric equations of the line of intersection of the planes $x - 2 y + 4 z = 2$ and $x + y - 2 z = 5$ are

A) $x = 4 , y = 2 t + 1 , z = - t$
B) $x = 2 , y = - 2 t + 1 , z = t + 1$
C) $x = 4 , y = 2 t + 1 , z = t - 1$
D) $x = 4 , y = 2 t + 1 , z = t$
E) $x = 2 , y = 2 t - 3 , z = t + 1$

Question

The parametric equations of the line of intersection of the planes $x - 2 y + 4 z = 2$ and $x + y - 2 z = 5$ are

A) $x = 4 , y = 2 t + 1 , z = - t$
B) $x = 2 , y = - 2 t + 1 , z = t + 1$
C) $x = 4 , y = 2 t + 1 , z = t - 1$
D) $x = 4 , y = 2 t + 1 , z = t$
E) $x = 2 , y = 2 t - 3 , z = t + 1$

Quizplus · Accepted Answer

To find the line of intersection of two planes, we can set up a system of equations based on the given plane equations and solve for two variables in terms of a parameter (let's say $t$). The given planes are $x - 2y + 4z = 2$ and $x + y - 2z = 5$. We can solve this system by various methods, such as elimination or substitution, but a common approach is to express two of the variables in terms of the third one, which will act as the parameter.One way to proceed is to eliminate one of the variables (for example, $x$) by adding or subtracting the equations, and then choose a variable to represent as a parameter (let's choose $z = t$). After solving, we find that the line of intersection can be represented parametrically as $x = 4, y = 2t + 1, z = t$, which matches option D. This process involves algebraic manipulation and choosing a parameter to express the line of intersection in a parametric form.

The Parametric Equations of the Line of Intersection of the Planes