Find the equation of tangent to the ellipse  from Mathematics

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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.


The equation of ellipse is straight x squared over straight a squared plus straight y squared over straight b squared space equals space 1
Differentiating both sides w.r.t.x,
                fraction numerator 2 straight x over denominator straight a squared end fraction plus fraction numerator 2 straight y over denominator straight b squared end fraction dy over dx space equals space 0 space space space space space space space space or space space space fraction numerator 2 straight y over denominator straight b squared end fraction dy over dx space equals space minus fraction numerator 2 straight x over denominator straight a squared end fraction

therefore space space space space dy over dx equals negative straight b squared over straight a squared straight x over straight y
space At space left parenthesis straight x subscript 1 comma space space straight y subscript 1 right parenthesis. space space dy over dx space equals space minus straight b squared over straight a squared straight x subscript 1 over straight y subscript 1 comma space which space is space slope space of space tangent
space therefore space space space space space the space equation space of space tangent space at space left parenthesis straight x subscript 1 comma space space straight y subscript 1 right parenthesis space is
space space space space space space space space space space space space space space straight y minus straight y subscript 1 space equals space minus straight b squared over straight a squared straight x subscript 1 over straight y subscript 1 left parenthesis straight x subscript 1 comma space space straight y subscript 1 right parenthesis space is
or space space space space straight y subscript 1 over straight b squared left parenthesis straight y minus straight y subscript 1 right parenthesis space equals space minus straight x subscript 1 over straight a squared left parenthesis straight x minus straight x subscript 1 right parenthesis space space space space or space space space space space yy subscript 1 over straight b squared minus straight y subscript 1 squared over straight a squared space equals space minus xx subscript 1 over straight a squared plus straight x subscript 1 squared over straight a squared
or space space space space space xx subscript 1 over straight a squared plus yy subscript 1 over straight b squared space equals space straight x subscript 1 squared over straight a squared plus straight y subscript 1 squared over straight b squared space space space space space space space space or space space space space space space space xx subscript 1 over straight a squared plus yy subscript 1 over straight b squared space equals space 1
                                                                        open square brackets table row cell because space space left parenthesis straight x subscript 1 comma space space straight y subscript 1 right parenthesis space lies space on space straight x squared over straight a squared plus straight y squared over straight b squared space equals space 1 end cell row cell therefore space straight x subscript 1 squared over straight a squared plus fraction numerator space straight y subscript 1 squared over denominator straight b squared end fraction equals space 1 end cell end table close square brackets

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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.

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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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