Exams Joint Entrance Examination Mathematics 3D Geometry Question 509 Question 509 of 521 Hard MCQ
The plane 2 x − y + z = 4 2x-y+z=4 2 x − y + z = 4 intersects the line segment joining the points A ( a , − 2 , 4 ) a,-2,4) a , − 2 , 4 ) and B ( 2 , b , − 3 ) 2,b,-3) 2 , b , − 3 ) at the point C in the ratio 2 : 1 and the distance of the point C from the origin is \[\sqrt{5}\]
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\[C:\left( {{{a + 4} \over 3},{{ - 2 + 2b} \over 3},{{4 - 6} \over 3}} \right)\]
\[ = \left( {{{a + 4} \over 3},{{ - 2 + 2b} \over 3},{{ - 2} \over 3}} \right)\]
C lies on plane 2 x − y + z = 4 2x - y + z = 4 2 x − y + z = 4
\[2\left( {{{a + 4} \over 3}} \right) - \left( {{{2b - 2} \over 3}} \right) - {2 \over 3} = 4\]
\[ \Rightarrow 2a + 8 - 2b + 2 - 2 = 12\]
\[ \Rightarrow a - b = 2\] ..... (i)
Now, \[OP = \sqrt 5 \]
\[{\left( {{{a + 4} \over 3}} \right)^2} + {\left( {{{2b - 2} \over 3}} \right)^2} + {4 \over 9} = 5\] and using (i)
\[a = {{11} \over 5},1\]
\[ \Rightarrow b = {1 \over 5}, - 1\]
as also \[ \Rightarrow a = 1,b = - 1\]
\[P(2, - 1, - 3),c\left( {{5 \over 3},{{ - 4} \over 3},{{ - 2} \over 3}} \right)\]
\[C{P^2} = {1 \over 9} + {1 \over 9} + {{49} \over 9} = {{17} \over 3}\] Answer the question to unlock