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Application Of Derivatives

Question
CBSEENMA12035161
Wired Faculty App

Find the equation of the tangent to the curve straight y equals square root of 3 straight x minus 2 end root which is parallel to the line 4x - 2y + 5 = 0

Solution

The equation of curve is  straight y space equals square root of 3 straight x minus 2 end root.                        ...(1)
therefore space space space space space dy over dx space equals fraction numerator 3 over denominator 2 square root of 3 straight x minus 2 end root end fraction therefore space space space space at space left parenthesis straight x comma space straight y right parenthesis comma space space slope space of space tangent space space equals space fraction numerator 3 over denominator 2 square root of 3 straight x minus 2 end root end fraction therefore space space space space space straight m subscript 1 space equals space fraction numerator 3 over denominator 2 square root of 3 straight x minus 2 end root end fraction comma space space space where space straight m subscript 1 space is space slope space of space tangent Let space space straight m subscript 2 space be space slope space of space line space 4 straight x minus 2 straight y plus 5 space equals space 0 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
therefore space space space space straight m subscript 2 space equals space minus fraction numerator 4 over denominator negative 2 end fraction space equals space 2 because space space space space space space tangent space is space parallel space to space line space left parenthesis 2 right parenthesis therefore space space space space space space space straight m subscript 1 space equals space straight m subscript 2 space space space rightwards double arrow space space space space space fraction numerator 3 over denominator 2 square root of 3 straight x minus 2 end root end fraction space equals space 2 space space space rightwards double arrow space 3 space equals space 4 space square root of 3 straight x minus 2 end root rightwards double arrow space space space space space space 9 space equals space 16 left parenthesis 3 straight x minus 2 right parenthesis space space space space space rightwards double arrow space space space space space 9 space equals space 48 straight x minus 32 space space space space space rightwards double arrow space space space space 48 straight x space equals space 41 space space space rightwards double arrow space space space space straight x space equals space 41 over 48 When space straight x space equals 41 over 48 comma space space space space space space straight y space equals space square root of 41 over 16 minus 2 end root space equals space square root of 9 over 16 end root space equals space 3 over 4 therefore space space space space space space space point space is space open parentheses 41 over 48 comma space space space 3 over 4 close parentheses therefore space space space space space equation space of space tangent space space is space space space space space space space space space space space space space space space space straight y minus 3 over 4 space space equals space 2 open parentheses straight x minus 41 over 48 close parentheses comma space space space space or space space space space space space straight y minus 3 over 4 space equals space 2 straight x minus 41 over 24 or space space space space space 24 straight y minus 18 space equals space 48 straight x minus 41 space space space space space or space space space space 48 straight x minus 24 straight y minus 23 space equals space 0

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