Application of Derivatives

Find the equations of normal lines to the curve y = x 2 – 3 x which are parallel to the line x + 9y = 14. - Wired Faculty

Question

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Find the equations of normal lines to the curve y = x² – 3 x which are parallel to the line x + 9y = 14.

Solution

The equations of curve is $straight y equals straight x cubed minus 3 straight x$ ...(1)
Differentiating w.r.t.x, $dy over dx space equals 3 straight x squared minus 3$
$therefore space space space space at space left parenthesis straight x comma space space straight y right parenthesis comma space space slope space of space tangent space space equals space 3 straight x squared minus 3 space space space rightwards double arrow space space space space space at space left parenthesis straight x comma space straight y right parenthesis comma space slope space of space normal space equals space minus fraction numerator 1 over denominator 3 straight x squared minus 3 end fraction therefore space space space space space space straight m subscript 1 space equals space minus fraction numerator 1 over denominator 3 straight x squared minus 3 end fraction Let space space straight m subscript 2 space be space slope space of space line space straight x plus 9 straight y space equals space 14 therefore space space space space space space straight m subscript 2 space equals space minus 1 over 9 therefore space space space space space space space space normal space is space parallel space to space left parenthesis 2 right parenthesis space space space space space therefore space space space straight m subscript 1 space equals space straight m subscript 2 therefore space space space space space space fraction numerator 1 over denominator 3 straight x squared minus 3 end fraction space equals space minus 1 over 9 space space space space space rightwards double arrow space space space space space space fraction numerator 1 over denominator straight x squared minus 1 end fraction space equals space 1 third rightwards double arrow space space space space space space straight x squared minus 1 space equals space 3 comma space space space space space rightwards double arrow space space space straight x squared space equals space 4 comma space space space space rightwards double arrow space space space space straight x space equals space 2 comma space space minus 2 When space straight x space equals 2 comma space space from space left parenthesis 1 right parenthesis comma space space straight y space equals space 8 minus 6 space equals space 2 therefore space space space points space is space left parenthesis 2 comma space space 2 right parenthesis therefore space space space space equation space of space normal space at space left parenthesis 2 comma space 2 right parenthesis space is space straight y minus 2 space equals space minus 1 over 9 left parenthesis straight x minus 2 right parenthesis comma space space space or space space 9 straight y minus 18 space equals space minus straight x plus 2 or space space space space straight x plus 9 straight y minus 20 space equals space 0 When space straight x space equals space minus 2 comma space from space left parenthesis 1 right parenthesis comma space straight y space equals negative 8 plus 6 space equals space minus 2 therefore space space space space space points space is space left parenthesis negative 2 comma space space space minus 2 right parenthesis therefore space space space space equation space of space normal space at space left parenthesis negative 2 comma space minus 2 right parenthesis space is space straight y minus left parenthesis negative 2 right parenthesis space equals space minus 1 over 9 left square bracket straight x minus left parenthesis negative 2 right parenthesis right square bracket or space space space space space space 9 straight y plus 18 space equals straight x minus 2 space space space space or space space straight x plus 9 straight y plus 20 space equals space 0.$

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