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

The equations of curve are
$straight x space equals sin space 3 space straight t comma space space space space straight y space equals space cos space 2 straight t$
At $straight t space equals space straight pi over 4 comma space space space space space space space straight x space equals space sin space fraction numerator 3 straight pi over denominator 4 end fraction space equals space sin space open parentheses straight pi minus straight pi over 4 close parentheses space equals space sin space straight pi over 4 space equals space fraction numerator 1 over denominator square root of 2 end fraction comma space space space straight y space equals space cos straight pi over 2 space equals space 0$
$therefore space space space space space space point space of space contact space is space open parentheses fraction numerator 1 over denominator square root of 2 end fraction comma space space 0 close parentheses space space space space space space dx over dt space equals space 3 space cos 3 straight t comma space space dy over dt space equals space minus 2 space sin space 2 straight t space space space space space dy over dx space equals space fraction numerator begin display style dy over dt end style over denominator begin display style dx over dt end style end fraction space equals space fraction numerator negative 2 space sin space 2 straight t over denominator 3 space cos space 3 straight t end fraction$
$At space straight t space equals space straight pi over 4 comma space space space dy over dx space equals space minus 2 over 3 fraction numerator sin space begin display style straight pi over 2 end style over denominator cos space begin display style fraction numerator 3 straight pi over denominator 4 end fraction end style end fraction space equals space minus 2 over 3 fraction numerator 1 over denominator cos space open parentheses straight pi minus begin display style straight pi over 4 end style close parentheses end fraction space equals space fraction numerator 2 over denominator 3 cos space begin display style straight pi over 4 end style end fraction space equals space fraction numerator 2 over denominator 3 cross times begin display style fraction numerator 1 over denominator square root of 2 end fraction end style end fraction space equals space fraction numerator 2 square root of 2 over denominator 3 end fraction$
$therefore space space space space slope space of space tangent space space equals space fraction numerator 2 square root of 2 over denominator 3 end fraction therefore space space space space the space equation space of space tangent space at space open parentheses fraction numerator 1 over denominator square root of 2 end fraction comma space 0 close parentheses space is space space space space space space space straight y minus 0 space equals space fraction numerator 2 square root of 2 over denominator 3 end fraction open parentheses straight x minus fraction numerator 1 over denominator square root of 2 end fraction close parentheses space space space space or space space space space space space 3 straight y space equals space 2 square root of 2 straight x minus 2 space space space space or space space 2 square root of 2 straight x minus 3 straight y minus 2 space equals space 0$

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

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