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Short Answer Type

91. Find the equation of the tangent and normal to the given curves at the points given:
y = x⁴ – 6x³ + 13x² – 10x + 5 at ( 0 , 5 )

The equations of curve is y = x⁴ – 6x³ + 13x² – 10x + 5
$therefore space space space space dy over dx space equals space 4 straight x cubed minus 3 bx squared plus 26 straight x minus 10$
At (0, 5). $dy over dx space equals space 0 minus 0 plus 0 minus 10 space equals space minus 10 comma space space which space is space slope space of space tangent$
$therefore space space space space space space equation space of space tangent space at thin space left parenthesis 0 comma space 5 right parenthesis space is space space space space space space space straight y space minus space 5 space space equals space minus 10 left parenthesis straight x minus 0 right parenthesis comma space space space space or space space space space straight y minus 5 space equals space minus 10 space straight x space space space space space or space space space 10 straight x plus straight y minus 5 space equals space 0 Slope space of space normal space equals space 1 over 10 therefore space space space space space space space equation space of space normal space at space left parenthesis 0 comma space 5 right parenthesis space is space space space space space space space space space space straight y minus 5 space equals space 1 over 10 left parenthesis straight x minus 0 right parenthesis or space space space space 10 straight y minus 50 space equals space straight x space space space space space space or space space space straight x minus 10 straight y plus 50 space equals space 0$

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92. Find the equation of the tangent and normal to the given curves at the points given:
y = x⁴ –6x³+ 13x²– 10x + 5 at (1 ,3)

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93. Find the equation of the tangent and normal to the given curves at the points given:

straight y squared space equals space fraction numerator straight x cubed over denominator 4 minus straight x end fraction space space at space space left parenthesis 2 comma space space minus 2 right parenthesis

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94. Find the equation of the tangent and normal to the given curves at the points given:

straight y space equals fraction numerator 5 straight x squared over denominator 1 plus straight x squared end fraction

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Long Answer Type

95. Find the equation of the tangent and normal to the given curves at the points given:

square root of straight x plus square root of straight y space equals space 5

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96. Find the equation of the tangent and normal to the given curves at the points given:

16 straight x squared plus 9 straight y squared space equals space 144 space space space at space left parenthesis straight x subscript 1 comma space space straight y subscript 1 right parenthesis space space space where space straight x subscript 1 space equals space 2 space space space and space straight y subscript 1 greater than 0

Also find the points of intersection where both tangent and normal cut the x-axis.

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97. Find the equation of the tangent and normal to the given curves at the points given:

square root of straight x plus square root of straight y space equals space straight a space space space at space space open parentheses straight a squared over 4 comma space space straight a squared over 4 close parentheses

Also find the points of intersection where both tangent and normal cut the x-axis.

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98. Find the equation of the tangent to the curve y = (x³ – 1) (x – 2) at points where the curve cuts the x-axis.

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Short Answer Type

99. Find the equation of the tangent to the curve

straight y equals space fraction numerator straight x minus 7 over denominator left parenthesis straight x minus 2 right parenthesis thin space left parenthesis straight x minus 3 right parenthesis end fraction

at the point where it cuts the x-axis.

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Long Answer Type

100.

Prove that the line $straight x over straight a plus straight y over straight b space equals space 1$ is a tangent to the curve $straight y space equals be to the power of negative straight x over straight a end exponent$ at the point where the curve cuts y-axis.