Find the equation of tangent and normal to the hyperbola  from

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 Multiple Choice QuestionsShort Answer Type

61.

Find the slope of the normal at the point (am3 , am2 ) to the curve a x2 = y3.

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62.

Find a point on the curve y = (x – 2)2 at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).

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63.

Find the point at which the tangent to the curve straight y equals space square root of 4 straight x minus 3 end root space minus 1 has its slope 2 over 3.

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64.

Find the equation of the normal to the curve y2 = 4x at the point (1, 2).

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65.

Find the equation of tangent to the ellipse straight x squared over straight a squared plus straight y squared over straight b squared space equals space 1 space space space at space left parenthesis straight x subscript 1 comma space space straight y subscript 1 right parenthesis.

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66.

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 at space space left parenthesis straight x subscript 0 comma space space straight y subscript 0 right parenthesis.


The equation of hyperbola is straight x squared over straight a squared minus straight y squared over straight b squared space equals 1
Differentiating both sides w.r.t. x,
                  fraction numerator 2 straight x over denominator straight a squared end fraction minus fraction numerator 2 straight y over denominator straight b squared end fraction dy over dx space equals space 0 comma space space space space space or space space space space fraction numerator 2 straight y over denominator straight b squared end fraction dy over dx space equals fraction numerator 2 straight x over denominator straight a squared end fraction space space space space rightwards double arrow space space space space dy over dx space equals space straight b squared over straight a squared straight x over straight y
At space left parenthesis straight x subscript 0 comma space space straight y subscript 0 right parenthesis comma space space space dy over dx space equals space straight b squared over straight a squared straight x subscript 0 over straight y subscript 0 comma space which space is space slope space of space tangent.
therefore space space space space space space space the space equation space of space tangent space at space left parenthesis straight x subscript 0 comma space straight y subscript 0 right parenthesis space is
space space space space space space space space space space space space space space space space space straight y minus straight y subscript 0 space equals space straight b squared over straight a squared straight x subscript 0 over straight y subscript 0 left parenthesis straight x minus straight x subscript 0 right parenthesis
or space space space space space space space straight y subscript 0 over straight b squared left parenthesis straight y minus straight y subscript 0 right parenthesis space equals space straight x subscript 0 over straight a squared left parenthesis straight x minus straight x subscript 0 right parenthesis space space space or space space space space space space yy subscript 0 over straight b squared minus straight y subscript 0 squared over straight b squared space equals space fraction numerator straight x space straight x subscript 0 over denominator straight a squared end fraction minus straight x subscript 0 squared over straight a squared
or space space space space space space fraction numerator straight x space straight x subscript 0 over denominator straight a squared end fraction minus fraction numerator straight y space straight y subscript 0 over denominator straight b squared end fraction space equals space straight x subscript 0 squared over straight a squared minus straight y subscript 0 squared over straight b squared
or space space space space space fraction numerator straight x space straight x subscript 0 over denominator straight a squared end fraction minus fraction numerator straight y space straight y subscript 0 over denominator straight b squared end fraction space equals space 1 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets table row cell because space space space space left parenthesis straight x subscript 0 comma space space straight y subscript 0 right parenthesis space lies space on space straight x squared over straight a squared minus straight y squared over straight b squared equals 1 end cell row cell therefore space space space straight x subscript 0 squared over straight a squared minus straight y subscript 0 squared over straight b squared space equals space 1 end cell end table close square brackets

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67. Find the equation of the normal at the point (am2, am3) for the curve ay= x3.
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68.

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 at space left parenthesis 1 comma space 1 right parenthesis

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69.

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

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 Multiple Choice QuestionsLong Answer Type

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