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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
$straight x space equals space cost comma space space space space space straight y space equals space sin space straight t space space space at space straight t space equals space straight pi over 4$

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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.

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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.

The equation of curve is y = x² + 4x + 1 ...(1)
When x = 3, y = 9 + 12 + 1 = 22
$therefore$ point with abscissa 3 is (3, 22)
Diff. (1) w.r.t.x, $dy over dx space equals space 2 straight x plus 4$
At (3, 22), $dy over dx space equals space 6 plus 4 space equals space 10$

$therefore space space space space space slope space of space tangent space space equals space 10 therefore space space space space space space equation space of space the space tangent space at space left parenthesis 3 comma space 22 right parenthesis space is space space space space space space space space space space space space space space space space space space straight y space minus 22 space equals space 10 left parenthesis straight x minus 3 right parenthesis or space space space space space space space space space space space straight y minus 22 space equals space 10 straight x minus 30 space space space space or space space space 10 straight x minus straight y minus 8 space equals space 0 Also space slope space of space normal space space equals space minus 1 over 10 therefore space space space equation space of space normal space at space left parenthesis 3 comma space 222 right parenthesis space is space space space space space space straight y minus 22 space equals space minus 1 over 10 left parenthesis straight x minus 3 right parenthesis space space space space or space space space space space space 10 straight y minus 220 space equals space minus straight x plus 3 space space space or space space space straight x plus 10 straight y minus 223 space equals space 0$