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CBSE Class 12-science Answered

Find the shortest distance of the point (0,c) from the parabola y=x2 ,where 0<c<5.
Asked by surendersial | 11 Mar, 2016, 10:12: PM
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begin mathsize 12px style Let space any space point space on space the space parabola space straight y equals straight x squared space be space open parentheses straight h comma space straight h squared close parentheses Distance space of space this space point space from space left parenthesis 0 comma space straight c right parenthesis space is straight S equals square root of open parentheses straight h minus 0 close parentheses squared plus open parentheses straight h squared minus straight c close parentheses squared end root rightwards double arrow straight S squared equals straight h squared plus straight h to the power of 4 plus straight c squared minus 2 straight h squared straight c rightwards double arrow straight S squared equals straight h to the power of 4 plus straight h squared open parentheses 1 minus 2 straight c close parentheses plus straight c squared straight S squared space is space the space variable space of space which space we space need space to space find space minima. FInd space the space first space derivative space with space respect space to space apostrophe straight h apostrophe 2 straight S dS over dh equals 4 straight h cubed plus 2 straight h open parentheses 1 minus 2 straight c close parentheses equals 0 rightwards double arrow straight h open parentheses 4 straight h squared plus 2 open parentheses 1 minus 2 straight c close parentheses close parentheses equals 0 rightwards double arrow straight h equals 0 space or space straight h equals plus-or-minus square root of fraction numerator 2 straight c minus 1 over denominator 2 end fraction end root When space 0 less than straight c less than 1 half comma space we space get space negative space value space for space straight h squared. space Hence comma space straight h equals 0 space gives space the space shortest space distance. space In space this space case space shortest space distance space straight S subscript min. end subscript equals square root of straight c squared end root equals straight c When space straight c greater than 1 half comma space the space first space derivative space changes space sign space from space minus ve space to space plus ve at space straight h equals plus-or-minus square root of fraction numerator 2 straight c minus 1 over denominator 2 end fraction end root. space hence comma space we space get space minima space there. In space this space case space straight S equals square root of straight h to the power of 4 plus straight h squared open parentheses 1 minus 2 straight c close parentheses plus straight c squared end root equals square root of open parentheses straight c minus 1 half close parentheses squared minus 2 open parentheses straight c minus 1 half close parentheses squared plus straight c squared end root equals square root of straight c squared minus open parentheses straight c minus 1 half close parentheses squared end root equals square root of 1 half cross times open parentheses 2 straight c minus 1 half close parentheses end root equals 1 half square root of 4 straight c minus 1 end root So space shortest space distance space is space apostrophe straight c space when space 0 less than straight c less than 1 half and space shortest space distance space is space 1 half square root of 4 straight c minus 1 end root comma space when space straight c greater than 1 half end style
Answered by satyajit samal | 14 Mar, 2016, 12:12: PM

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CBSE 12-science - Maths
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