logo

Ask Questions, Get Answers

 
X
 Search
Want to ask us a question? Click here
Browse Questions
Ad
Home  >>  CBSE XII  >>  Math  >>  Model Papers
0 votes

Find the distance of the point (2, 12, 5) from the point of intersection of the line $ \overrightarrow r = 2\hat i - 4\hat j + 2 \hat k + \lambda ( 3\hat i + 4 \hat j + 2 \hat k ) $ and the plane $ \overrightarrow r. ( \hat i - 2 \hat j + \hat k ) =0.$

$(a)\;13\;units\qquad(b)\;1\;units\qquad(c)\;12\;units\qquad(d)\;14\;units$

Can you answer this question?
 
 

1 Answer

0 votes
Equation of the given line is
$\overrightarrow r = 2\hat i - 4\hat j + 2 \hat k + \lambda ( 3\hat i + 4 \hat j + 2 \hat k ) $ -----------(1)
Given equation of the plane is
$ \overrightarrow r = ( \hat i - 2 \hat j+ \hat k ) = 0 $--------(2)
Substituting the value of $ \overrightarrow r $ in eqn 2 we get
$( 2\hat i - 4\hat j + 2 \hat k) + \lambda ( 3\hat i + 4 \hat j + 2 \hat k ) = ( \hat i - 2 \hat j + \hat k ) = 0$
$ \Rightarrow [ (2+3 \lambda) \hat i + (4 \lambda - 4 ) \hat j + (2+2 \lambda) \hat k ].[ \hat i - 2 \hat j + \hat k ] = 0$
$ \Rightarrow (2+ 3 \lambda)(1)+(4 \lambda-4)(-2)+(2+2 \lambda)(1)=0$
$ 2+ 3 \lambda-8\lambda+8+2+2 \lambda = 0$ Hence the equation of the line is
$\overrightarrow r = (2\hat i - 4\hat j + 2 \hat k) + 4 ( 3\hat i + 4 \hat j + 2 \hat k ) $
(i.e.,) $ \overrightarrow r = 2\hat i - 4\hat j + 2 \hat k +12 \hat i + 16 \hat j + 8 \hat k $
$ = 14 \hat i + 18 \hat k + 10 \hat k$
The point of intersection of the given line and the plane is given by the co-ordinate
(14, 12, 10), the given point is ( 2, 12, 5)
hence the distance between the point is
$ d = \sqrt{ (14-2)^2 + (12-12)^2+ (10-5)^2}$
=$ \sqrt{ 12^2+0+5^2}$
=$ \sqrt{144+25}$
=$ \sqrt{144+25}$
=$ \sqrt{169}$
$= 13$
hence the distance is 13 units
answered Mar 31, 2014 by thanvigandhi_1
edited Apr 1, 2014 by thanvigandhi_1
 

Related questions

Ask Question
student study plans
x
JEE MAIN, CBSE, NEET Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App
...