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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.

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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:
$straight x space equals space straight theta plus space sin space straight theta comma space space space straight y space equals space 1 plus space cos space straight theta space space at space straight theta equals space straight pi over 4$

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75.

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

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76.

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

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77.

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

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Short Answer Type

78.

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

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79. Find the equation of the tangent and normal lines to the curve
$straight y space equals sin squared straight x space at space space space straight x space equals space straight pi over 2.$

The equation of curve is $straight y space equals space sin squared straight x$ ...(1)
$therefore space space space space dy over dx space equals space 2 space sinx space cosx space space equals space sin space 2 straight x$
$At space space straight x space equals space straight pi over 2 comma space space space dy over dx space equals space sin space straight pi space equals space 0 therefore space space space space slope space of space tangent space space equals space 0 When space straight x space equals space straight pi over 2 comma space space space space space space space straight y space equals space sin squared straight pi over 2 space equals space 1 therefore space space space point space is space open parentheses straight pi over 2 comma space space 1 close parentheses The space equation space of space tangent space at space open parentheses straight pi over 2 comma space space 1 close parentheses space is space space space space space space space space space space space space space space space space space space space space space space space space space straight y space minus space 1 space equals space 0 space open parentheses straight x minus straight pi over 2 close parentheses space space space or space space space space straight y minus 1 space equals space 0$
Since tangent is parallel to x-axis.
$therefore space space space space space normal space is space parallel space to space straight y minus axis therefore space space space space space space equation space of space normal space through space open parentheses straight pi over 2 comma space space 1 close parentheses space is space space space space space space space space space space space space space straight x space equals space straight pi over 2 space space space space space or space space space space space straight x minus straight pi over 2 space equals space 0.$

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80.

Find the equations of the tangent and normal to the curve y = x² + 4x + 1 at the point whose abscissa is 3.

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