Question:
A line with positive direction cosines passes through the point $P (2, -1, 2)$ and makes equal angles with the coordinate axes. The line meets the plane $2x + y+ z = 9$ at point $Q$. The length of the line segment $PQ$ equals
A line with positive direction cosines passes through the point $P (2, -1, 2)$ and makes equal angles with the coordinate axes. The line meets the plane $2x + y+ z = 9$ at point $Q$. The length of the line segment $PQ$ equals
Updated On: Jun 14, 2022
- $1$
- $\sqrt 2$
- $\sqrt 3$
- $2$
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The Correct Option is C
Solution and Explanation
Since, $\hspace15mm l = m = n \frac{1}{\sqrt 3}$
$\therefore$ Equations of line are $\frac{x-2}{1/ \sqrt 3 } = \frac{y+1}{1/ \sqrt 3 } = \frac{z-2}{1/ \sqrt 3 }$
$\Rightarrow $$\hspace40mm x - 2 = y +1 = z - 2 = r \, \, \, $ [say]
$\therefore$ Any point on the line is
$\hspace30mm Q = (r+2,r-1,r+2)$
$ \because Q $ lies on the plane $ 2x+ y +z = 9 $
$\therefore$ $\hspace5mm 2(r+2)+(r-1)+(r+2) = 9$
$\Rightarrow$ $\hspace40mm 4r + 5 = 9$
$\Rightarrow$ $\hspace50mm r = 1 $
$\Rightarrow$ $\hspace15mm Q(3,\, 0,\, 3)$
$\therefore$ $ PQ= {\sqrt{(3-2)^2+(0+1)^2+(3-2)^2}} = \sqrt 3$
$\therefore$ Equations of line are $\frac{x-2}{1/ \sqrt 3 } = \frac{y+1}{1/ \sqrt 3 } = \frac{z-2}{1/ \sqrt 3 }$
$\Rightarrow $$\hspace40mm x - 2 = y +1 = z - 2 = r \, \, \, $ [say]
$\therefore$ Any point on the line is
$\hspace30mm Q = (r+2,r-1,r+2)$
$ \because Q $ lies on the plane $ 2x+ y +z = 9 $
$\therefore$ $\hspace5mm 2(r+2)+(r-1)+(r+2) = 9$
$\Rightarrow$ $\hspace40mm 4r + 5 = 9$
$\Rightarrow$ $\hspace50mm r = 1 $
$\Rightarrow$ $\hspace15mm Q(3,\, 0,\, 3)$
$\therefore$ $ PQ= {\sqrt{(3-2)^2+(0+1)^2+(3-2)^2}} = \sqrt 3$
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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.
