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Vector Algebra

Question
CBSEENMA12034051
Wired Faculty App

Points E and F are taken on the sides BC and CD of a parallelogram ABCD such that open vertical bar BF with rightwards arrow on top close vertical bar thin space colon thin space open vertical bar FC with rightwards arrow on top close vertical bar space equals space straight mu comma space space space space open vertical bar DE with rightwards arrow on top close vertical bar space colon thin space open vertical bar EC with rightwards arrow on top close vertical bar space equals space straight lambda. The straight lines FD and AE intersect at the point O. Find the ratio open vertical bar FO with rightwards arrow on top close vertical bar space colon thin space open vertical bar OD with rightwards arrow on top close vertical bar.

Solution
Take A as origin. Let straight b with rightwards arrow on top and straight d with rightwards arrow on top be position vectors of B and D respectively.
Now,     AC with rightwards arrow on top space equals space AB with rightwards arrow on top space plus space BC with rightwards arrow on top
 therefore space space AC with rightwards arrow on top space equals space AB with rightwards arrow on top space plus space AD with rightwards arrow on top therefore space space space AC with rightwards arrow on top space equals space straight b with rightwards arrow on top plus straight d with rightwards arrow on top therefore space space space space straight P. straight V. space of space straight C space is space left parenthesis straight b with rightwards arrow on top plus straight d with rightwards arrow on top right parenthesis straight F space divides space BC space in space the space ratio space straight mu space colon space 1. therefore space space space space space straight F space is space open parentheses fraction numerator straight mu space left parenthesis straight b with rightwards arrow on top plus straight d with rightwards arrow on top right parenthesis space plus space 1. space straight b with rightwards arrow on top over denominator straight mu plus 1 end fraction close parentheses space space space space space space space space space straight i. straight e. space open parentheses fraction numerator left parenthesis straight mu plus 1 right parenthesis space straight b with rightwards arrow on top space plus space straight mu space straight d with rightwards arrow on top over denominator straight mu plus 1 end fraction close parentheses Again space straight E space divides space DC space in space the space ratio space straight lambda space colon 1 space. therefore space straight E space is space open parentheses fraction numerator straight lambda left parenthesis straight b with rightwards arrow on top plus straight d with rightwards arrow on top right parenthesis space plus space 1. space straight d with rightwards arrow on top over denominator straight lambda plus 1 end fraction close parentheses space space straight i. straight e. space space open parentheses fraction numerator straight lambda straight b with rightwards arrow on top space plus space left parenthesis straight lambda plus 1 right parenthesis space straight d with rightwards arrow on top over denominator straight lambda plus 1 end fraction close parentheses Now space FD space and space AE space intersect space at space straight O. Let space DO space colon space OF space space equals space straight k subscript 1 colon 1 space space space and space AO thin space colon thin space OE space equals space straight k subscript 2 colon 1
therefore space space space fraction numerator straight k subscript 1 open parentheses begin display style fraction numerator left parenthesis straight mu plus 1 right parenthesis space straight b with rightwards arrow on top space plus space straight mu space straight d with rightwards arrow on top over denominator straight mu plus 1 end fraction end style close parentheses plus 1. space straight d with rightwards arrow on top over denominator straight k subscript 1 plus 1 end fraction space equals space fraction numerator straight k subscript 2 open parentheses begin display style fraction numerator left parenthesis straight lambda plus 1 right parenthesis space straight d with rightwards arrow on top plus space straight lambda space straight b with rightwards arrow on top over denominator straight lambda plus 1 end fraction end style close parentheses plus 1. space 0 with rightwards arrow on top over denominator straight k subscript 2 plus 1 end fraction therefore space space space fraction numerator straight k subscript 1 left parenthesis straight mu plus 1 right parenthesis space straight b with rightwards arrow on top space plus space left parenthesis straight k subscript 1 straight mu space plus space straight mu plus 1 right parenthesis space straight d with rightwards arrow on top over denominator left parenthesis straight k subscript 1 plus 1 right parenthesis thin space left parenthesis straight mu plus 1 right parenthesis end fraction space equals fraction numerator straight k subscript 2 left parenthesis straight lambda plus 1 right parenthesis space straight d with rightwards arrow on top space plus space straight k subscript 2 space straight lambda space straight b with rightwards arrow on top over denominator left parenthesis straight k subscript 2 plus 1 right parenthesis thin space left parenthesis straight lambda plus 1 right parenthesis end fraction therefore space space space fraction numerator straight k subscript 1 left parenthesis straight mu plus 1 right parenthesis over denominator left parenthesis straight k subscript 1 plus 1 right parenthesis thin space left parenthesis straight mu plus 1 right parenthesis end fraction straight b with rightwards arrow on top space plus space fraction numerator left parenthesis straight k subscript 1 space straight mu space plus space straight mu space plus 1 right parenthesis over denominator left parenthesis straight k subscript 1 plus 1 right parenthesis thin space left parenthesis straight mu plus 1 right parenthesis end fraction straight d with rightwards arrow on top space equals fraction numerator straight k subscript 2 straight lambda over denominator left parenthesis straight k subscript 2 plus 1 right parenthesis thin space left parenthesis straight lambda plus 1 right parenthesis end fraction straight d with rightwards arrow on top space plus space fraction numerator straight k subscript 2 left parenthesis straight lambda plus 1 right parenthesis over denominator left parenthesis straight k subscript 2 plus 1 right parenthesis thin space left parenthesis straight lambda plus 1 right parenthesis end fraction straight b with rightwards arrow on top rightwards double arrow space space space space space space space space space space space space space space space space space space space fraction numerator straight k subscript 1 left parenthesis straight mu plus 1 right parenthesis over denominator left parenthesis straight k subscript 1 plus 1 right parenthesis thin space left parenthesis straight mu plus 1 right parenthesis end fraction space equals fraction numerator straight k subscript 2 straight lambda over denominator left parenthesis straight k subscript 2 plus 1 right parenthesis space left parenthesis straight lambda plus 1 right parenthesis end fraction or space space space space space space space space space space space space space space space space space fraction numerator straight k subscript 1 over denominator straight k subscript 1 plus 1 end fraction space equals space fraction numerator straight k subscript 2 straight lambda over denominator left parenthesis straight k subscript 2 plus 1 right parenthesis thin space left parenthesis straight lambda plus 1 right parenthesis end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis and space space space fraction numerator straight k subscript 1 straight mu plus straight mu plus 1 over denominator left parenthesis straight k subscript 1 plus 1 right parenthesis thin space left parenthesis straight mu plus 1 right parenthesis end fraction space equals fraction numerator straight k subscript 2 over denominator straight k subscript 2 plus 1 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis

rightwards double arrow space space space space space space space space space space space space space space space straight k subscript 1 space equals space fraction numerator straight k subscript 1 straight mu space plus space straight mu space plus 1 over denominator straight mu plus 1 end fraction. space fraction numerator straight lambda over denominator straight lambda plus 1 end fraction rightwards double arrow space space space space space space space space space space space space space space space space straight k subscript 1 space equals space fraction numerator straight k subscript 1 space straight lambda space straight mu over denominator left parenthesis straight lambda plus 1 right parenthesis thin space left parenthesis straight mu plus 1 right parenthesis end fraction plus fraction numerator straight lambda left parenthesis straight mu plus 1 right parenthesis over denominator left parenthesis straight lambda plus 1 right parenthesis thin space left parenthesis straight mu plus 1 right parenthesis end fraction rightwards double arrow space space space space space straight k subscript 1 open square brackets 1 minus fraction numerator straight lambda space straight mu over denominator left parenthesis straight lambda plus 1 right parenthesis thin space left parenthesis straight mu plus 1 right parenthesis end fraction close square brackets space equals space fraction numerator straight lambda space left parenthesis straight mu plus 1 right parenthesis over denominator left parenthesis straight lambda plus 1 right parenthesis thin space left parenthesis straight mu plus 1 right parenthesis end fraction rightwards double arrow space space space straight k subscript 1 open square brackets fraction numerator left parenthesis straight lambda plus 1 right parenthesis thin space left parenthesis straight mu plus 1 right parenthesis space minus space λμ over denominator left parenthesis straight lambda plus 1 right parenthesis thin space left parenthesis straight mu plus 1 right parenthesis end fraction close square brackets space equals space fraction numerator straight lambda left parenthesis straight mu plus 1 right parenthesis over denominator left parenthesis straight lambda plus 1 right parenthesis thin space left parenthesis straight mu plus 1 right parenthesis end fraction rightwards double arrow space space space straight k subscript 1 fraction numerator straight lambda plus straight mu plus 1 over denominator left parenthesis straight lambda plus 1 right parenthesis thin space left parenthesis straight mu plus 1 right parenthesis end fraction space equals space fraction numerator straight lambda left parenthesis straight mu plus 1 right parenthesis over denominator left parenthesis straight lambda plus 1 right parenthesis thin space left parenthesis straight mu plus 1 right parenthesis end fraction rightwards double arrow space space space straight k subscript 1 left parenthesis straight lambda plus straight mu plus 1 right parenthesis space equals space straight lambda left parenthesis straight mu plus 1 right parenthesis space space space space space space space space rightwards double arrow space space space space space straight k subscript 1 space equals space fraction numerator straight lambda space left parenthesis straight mu plus 1 right parenthesis over denominator straight lambda plus straight mu plus 1 end fraction therefore space space space space space open vertical bar FO with rightwards arrow on top close vertical bar space colon space open vertical bar OD with rightwards arrow on top close vertical bar space equals space fraction numerator straight lambda plus straight mu plus 1 over denominator straight lambda left parenthesis straight mu plus 1 right parenthesis end fraction

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