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Short Answer Type

81.
Show that the normal at any point 0 to the curve
x = a cosө + a ө sinө. y = a sin ө – a ө cos ө
is at a constant distance from the origin.

The equation of the curve are

$straight x equals straight a space cos space straight theta space plus space straight a space straight theta space sin space straight theta comma space space space space space straight y equals straight a space sin space straight theta space minus space straight a space straight theta space cos space straight theta therefore space space space space dx over dθ space equals space minus straight a space sin space straight theta space space plus space straight a space straight theta space cos space straight theta space plus space straight a space sin space straight theta space equals space straight a space straight theta space cos space straight theta space space space space space space space dy over dθ space equals space straight a space cos space straight theta space plus space straight a space straight theta space sin space straight theta space minus space straight a space cos space straight theta space equals space straight a space straight theta space sin space straight theta therefore space space space space dy over dx space equals space fraction numerator begin display style dy over dθ end style over denominator begin display style dx over dθ end style end fraction space equals space fraction numerator straight a space straight theta space sin space straight theta over denominator straight a space straight theta space cos space straight theta end fraction space equals space tan space straight theta comma space space which space is space slope space of space tangent therefore space space space space space slope space of space normal space space equals space minus fraction numerator 1 over denominator tan space straight theta end fraction space equals space minus fraction numerator cos space straight theta over denominator sin space straight theta end fraction comma The space equation space of space normal space at space apostrophe straight theta apostrophe space is space space space space space space space straight y minus left parenthesis straight a space sin space straight theta space minus space space straight a space straight theta space cos space straight theta right parenthesis space equals space minus fraction numerator cos space straight theta over denominator sin space straight theta end fraction left square bracket straight x minus left parenthesis straight a space cosθ space plus space straight a space straight theta space sin space straight theta right parenthesis right square bracket space or space space space space straight y space sin space straight theta space minus space straight a space sin squared straight theta space plus space straight a space straight theta space sin space straight theta space cos space straight theta space equals space minus straight x space cos space straight theta space plus space straight a space cos squared straight theta space plus space straight a space straight theta space sin space straight theta space cos space straight theta or space space space space space space straight x space cosθ space plus space straight y space sin space straight theta space minus space straight a left parenthesis sin squared straight theta space plus space cos squared straight theta right parenthesis space equals space 0 or space space space space space space space straight x space cos space straight theta space plus space straight y space sin space straight theta space space minus straight a space equals space 0 Its space distance space from space origin space space equals space fraction numerator open vertical bar 0 left parenthesis cos space straight theta right parenthesis space plus space 0 left parenthesis sin space straight theta right parenthesis space minus straight a close vertical bar over denominator square root of cos squared straight theta space plus space sin squared straight theta end root end fraction space equals space straight a over 1 space equals space straight a comma space which space is space constant.$

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82.

Find the equation of the tangent and normal to the given curves at the points given:
y = x² at (0,0).

76 Views

83.

Find the equation of the tangent and normal to the given curves at the points given:
y² = 4ax at (0, 0).

89 Views

84.

Find the equation of the tangent and normal to the given curves at the points given:
y = x³ at (1, 1)

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85.

Find the equation of the tangent and normal to the given curves at the points given:
y = x² at (2, 8)

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Long Answer Type

86.

Find the equation of tangent and normal to the hyperbola $straight x squared over straight a squared minus straight y squared over straight b squared space equals space 1 space space space space at space the space point space space left parenthesis straight x subscript 0 comma space space space straight y subscript 0 right parenthesis$

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Short Answer Type

87. Find the equation of the normal at the point (am², am³) for the curve ay²= x³.

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88.

Find the equation of the tangent and the normal to the curve $straight x to the power of 2 over 3 end exponent plus straight y to the power of 2 over 3 end exponent space equals space 2 space space space at space space left parenthesis 1 comma space 1 right parenthesis.$

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89. Find the equation of the tangent and normal to the curve y² = 4ax at (at², 2at)

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Long Answer Type

90.

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