Find the equation of the tangent and normal to the curve from M

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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.

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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

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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


The equations of the curve are       straight x space equals space 1 minus space cos space straight theta comma space space space straight y space equals space straight theta minus sin space straight theta
At space straight theta space equals straight pi over 4 comma space space space space straight x space equals space 1 minus cos space straight pi over 4 space equals space 1 minus fraction numerator 1 over denominator square root of 2 end fraction comma space space space space space space space straight y space equals space straight theta minus sinθ space equals straight pi over 4 minus sin space straight pi over 4 space equals space straight pi over 4 minus fraction numerator 1 over denominator square root of 2 end fraction
therefore space space space space space space point space of space contact space is space open parentheses 1 minus fraction numerator 1 over denominator square root of 2 end fraction comma space space space space space straight pi over 4 minus fraction numerator 1 over denominator square root of 2 end fraction close parentheses
Also comma space space space space space dx over dθ space equals space sin space straight theta comma space space space space dy over dθ space equals space 1 minus space cos space straight theta
therefore space space space space space space space space space space dy over dx space equals space fraction numerator begin display style dy over dx end style over denominator begin display style dx over dθ end style end fraction space equals space fraction numerator 1 minus cos space straight theta over denominator sin space straight theta end fraction
At space straight theta space equals straight pi over 4 comma space space space space space space space dy over dx equals space fraction numerator 1 minus cos begin display style straight pi over 4 end style over denominator sin space begin display style straight pi over 4 end style end fraction space equals space fraction numerator 1 minus begin display style fraction numerator 1 over denominator square root of 2 end fraction end style over denominator square root of 2 end fraction space equals space square root of 2 minus 1
therefore space space space space space at space straight theta space equals space straight pi over 4 comma space space space slope space of space tangent space equals space square root of 2 minus 1
therefore space space space equation space of space tangent space is space space straight y minus open parentheses straight pi over 4 minus fraction numerator 1 over denominator square root of 2 end fraction close parentheses space equals space left parenthesis square root of 2 minus 1 right parenthesis space open square brackets straight x minus open parentheses 1 minus fraction numerator 1 over denominator square root of 2 end fraction close parentheses close square brackets
Slope space of space tangent space equals space square root of 2 minus 1
therefore space space space slope space of space normal space space equals space minus fraction numerator 1 over denominator square root of 2 minus 1 end fraction space equals space minus fraction numerator 1 over denominator square root of 2 minus 1 end fraction cross times fraction numerator square root of 2 plus 1 over denominator square root of 2 plus 1 end fraction equals negative fraction numerator square root of 2 plus 1 over denominator 2 minus 1 end fraction space equals negative left parenthesis square root of 2 plus 1 right parenthesis

therefore space space space equation space of space normal space is
space space space space space space space straight y minus open parentheses straight pi over 4 minus fraction numerator 1 over denominator square root of 2 end fraction close parentheses space equals space minus left parenthesis square root of 2 plus 1 right parenthesis space open square brackets straight x minus open parentheses 1 minus fraction numerator 1 over denominator square root of 2 end fraction close parentheses close square brackets

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