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Application of Derivatives

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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.

78 Views

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)

The equation of curve is y = x^{3
$therefore space space space space space dy over dx space equals space 3 straight x squared$}At (1, 1), $dy over dx space equals space 3 left parenthesis 1 right parenthesis squared space equals space 3$

$therefore space space space space straight m space equals space 3 comma space space where space straight m space is space slope space of space tangent. therefore space space space space space the space equation space of space tangent space at space left parenthesis 1 comma space 1 right parenthesis space is space space space space space space space space space space straight y minus 1 space equals space 3 left parenthesis straight x minus 1 right parenthesis semicolon space space space space space space or space space space straight y minus 1 space equals space 3 straight x minus 3 space space space space or space space space 3 straight x minus straight y minus 2 space equals space 0 Slope space of space normal space space equals space 1 third therefore space space space space equation space of space normal space at space left parenthesis 1 comma space 1 right parenthesis space is space space space space space space space space space space space space space space straight y minus 1 space equals space minus 1 third left parenthesis straight x minus 1 right parenthesis comma space space space space or space space space 3 straight y minus 3 space equals space minus straight x plus 1 space space space space or space space straight x plus 3 straight y minus 4 space equals space 0$

74 Views

85.

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

72 Views

Long Answer Type

86.

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

80 Views

Short Answer Type

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

83 Views

88.

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

84 Views

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

78 Views

Long Answer Type

90.

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