The equation of the plane containing the lines $2x - 5y + z = 3, x + y + \, z = 5$ and parallel to the plane $x + 3 y+ 6 z = 1$ is
- $2x + 6y + 12z = 13$
- $x + 3y + 6z = - 7$
\(x + 3y + 6z = 27\)
- $2x + 6y + 12x = - 13$
The Correct Option is C
Solution and Explanation
The correct option is(C): x+3y+6z=27.
The given lines are:
- 2x - 5y + z = 3
- x + y + z = 5
A(x−x0)+B(y−y0)+C(z−z0)=0,
where (A, B, C) is the direction vector of the plane, and (x_0, y_0, z_0) is a point on the plane.
Plugging in the values, we get:
1(x−3)+3(y+2)+6(z−4)=0,
which simplifies to:
x+3y+6z=27.
So, the equation of the plane containing the lines 2x - 5y + z = 3 and x + y + z = 5, and parallel to the plane x + 3y + 6z = 1, is
x+3y+6z=27.
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Concepts Used:
Three Dimensional Geometry
Mathematically, Geometry is one of the most important topics. The concepts of Geometry are derived w.r.t. the planes. So, Geometry is divided into three major categories based on its dimensions which are one-dimensional geometry, two-dimensional geometry, and three-dimensional geometry.
Direction Cosines and Direction Ratios of Line:
Consider a line L that is passing through the three-dimensional plane. Now, x,y and z are the axes of the plane and α,β, and γ are the three angles the line makes with these axes. These are commonly known as the direction angles of the plane. So, appropriately, we can say that cosα, cosβ, and cosγ are the direction cosines of the given line L.
