Question:
A variable plane passes through a fixed point (3, 2, 1) and meets x, y and z axes at A, B and C respectively. A plane is drawn parallel to yz-plane through A, a second plane is drawn parallel zx-plane through B and a third plane is drawn parallel to xy-plane through C. Then the locus of the point of intersection of these three planes, is :
A variable plane passes through a fixed point (3, 2, 1) and meets x, y and z axes at A, B and C respectively. A plane is drawn parallel to yz-plane through A, a second plane is drawn parallel zx-plane through B and a third plane is drawn parallel to xy-plane through C. Then the locus of the point of intersection of these three planes, is :
Updated On: Oct 1, 2024
- $\frac{x}{3} + \frac{y}{2} + \frac{z}{1} = 1 $
- $ x + y + z = 6 $
- $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{11}{6} $
- $\frac{3}{x} + \frac{2}{y} + \frac{1}{z} = 1 $
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The Correct Option is D
Solution and Explanation
Let plane is $\frac{x}{a}+\frac{y}{b}+\frac{z}{c} =1$
it passes through (3,2,1) $\therefore \frac{3}{a}+\frac{2}{b}+\frac{1}{c} = 1$
Now A (a,0,0), B (0, b, 0), C (0,0,c)
$\therefore$ Locus of point of intersection of planes x = a
y = b, z = c is $\frac{3}{x}+\frac{2}{y}+\frac{1}{z} = 1$
it passes through (3,2,1) $\therefore \frac{3}{a}+\frac{2}{b}+\frac{1}{c} = 1$
Now A (a,0,0), B (0, b, 0), C (0,0,c)
$\therefore$ Locus of point of intersection of planes x = a
y = b, z = c is $\frac{3}{x}+\frac{2}{y}+\frac{1}{z} = 1$
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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.
