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Application of Derivatives

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Short Answer Type

71. Find the equation of the tangent and normal to the curve x = a t² , y = 2 at at the point ‘t’.

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Long Answer Type

72. Find the equation of the tangent at

straight t space equals space straight pi over 4 space to space the space curve space straight x space equals sin space 3 straight t comma space space space space space straight y space equals space cos 2 straight t.

73 Views

Short Answer Type

73.

Find the equation of tangent to the curve given by x = a sin³ t, y = b cos³ t at a point where $straight t equals space straight pi over 2.$

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Long Answer Type

74.

Find the equation of tangent to the curve:
$straight x space equals space straight theta plus space sin space straight theta comma space space space straight y space equals space 1 plus space cos space straight theta space space at space straight theta equals space straight pi over 4$

73 Views

75.

Find the equation of tangent to the curve:
$straight x space equals space 1 minus cos space straight theta comma space space space straight y space equals space straight theta space plus space sin space straight theta space space at space straight theta space equals space straight pi over 4.$

74 Views

76.

Find the equation of tangent to the curve:
$straight x space equals space 1 plus cos space straight theta comma space space space space straight y equals space straight theta plus sinθ space space at space straight theta space equals space straight pi over 4.$

72 Views

77.

Find the equation of the tangent and normal to the curve
$straight x space equals space cost comma space space space space space straight y space equals space sin space straight t space space space at space straight t space equals space straight pi over 4$

78 Views

Short Answer Type

78.
Find the equation of the tangent to the curve
$straight y space equals space fraction numerator 1 plus sinx over denominator cosx end fraction space at space straight x space equals space straight pi over 4.$

The equation of curve is
$straight y space equals space fraction numerator 1 plus sinx over denominator cosx end fraction space equals space 1 over cosx plus sinx over cosx space equals space secx plus tanx$

$At space straight x space equals space straight pi over 4 comma space space space straight y space equals sec straight pi over 4 plus tan straight pi over 4 space equals space square root of 2 plus 1 therefore space space space space points space is space open parentheses straight pi over 4 comma space space square root of 2 plus 1 close parentheses space space space space space space space space space space space space space space dy over dx space equals space sec space straight x space tanx space plus sec squared straight x space equals space secx left parenthesis tanx space plus sec space straight x right parenthesis At space space space straight x space equals straight pi over 4 comma space space dy over dx space equals space sec straight pi over 4 open parentheses tan space straight pi over 4 space plus space sec space straight pi over 4 close parentheses space equals space square root of 2 left parenthesis 1 plus square root of 2 right parenthesis therefore space space space space equation space of space tangent space at space open parentheses straight pi over 4 comma space space space square root of 2 plus 1 close parentheses space space space is space space space space space space space space space space space space space space space straight y minus left parenthesis square root of 2 plus 1 right parenthesis space equals space square root of 2 left parenthesis 1 plus square root of 2 right parenthesis space open parentheses straight x minus straight pi over 4 close parentheses or space space space space space space space space space space straight y minus left parenthesis square root of 2 plus 1 right parenthesis space equals space square root of 2 left parenthesis 1 plus square root of 2 right parenthesis straight x minus fraction numerator square root of 2 left parenthesis 1 plus square root of 2 right parenthesis straight pi over denominator 4 end fraction space or space space space space space space space space space space 4 square root of 2 left parenthesis 1 plus square root of 2 right parenthesis straight x minus 4 straight y plus 4 left parenthesis square root of 2 plus 1 right parenthesis minus square root of 2 left parenthesis 1 plus square root of 2 right parenthesis space straight pi space equals space 0 space$