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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’.

The equations of curve are x = at² , y = 2at
$therefore space space space dx over dt space equals space 2 at comma space space dy over dt equals 2 straight a space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space space space space space dy over dx space equals space fraction numerator begin display style dy over dt end style over denominator begin display style dx over dt end style end fraction equals space fraction numerator 2 straight a over denominator 2 at end fraction space equals 1 over straight t$
$therefore space space space space slope space of space tangent space space equals space 1 over straight t$
$therefore space space space space space the space equation space of space tangent space at space left parenthesis at squared comma space space 2 at right parenthesis space is space space space space space space space space space space space space space space straight y minus 2 at space equals space 1 over straight t left parenthesis straight x minus at squared right parenthesis space space space space space or space space space space ty minus 2 at squared space equals space straight x minus at squared space space space space or space space space space straight x minus ty plus at squared space equals space 0 Also space slope space of space normal space space equals space minus fraction numerator 1 over denominator begin display style 1 over straight t end style end fraction space equals space minus straight t therefore space space space space space equation space of space normal space at space left parenthesis at squared comma space space space 2 at right parenthesis space is space space space space space space space space space space space space space space space space space space straight y minus 2 at space equals space minus straight t left parenthesis straight x minus at squared right parenthesis space or space space space space space space space space space space straight y minus 2 at space equals space minus tx plus at cubed space space space space space space space space space space space or space space tx plus straight y minus 2 at minus at cubed space equals space 0$

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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 $straight t equals space straight pi over 2.$

102 Views

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
$straight y space equals space fraction numerator 1 plus sinx over denominator cosx end fraction space at space straight x space equals space straight pi over 4.$